Abstract Algebra- Theory and Applications, 2016a
Abstract Algebra- Theory and Applications, 2016a
Abstract Algebra- Theory and Applications, 2016a
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1<br />
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<br />
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<br />
S <br />
S S <br />
<br />
<br />
<br />
<br />
<br />
3+56− 13 + 8/2<br />
<br />
2+3=5<br />
2x =6 x =4<br />
ax 2 + bx + c =0 a ≠0 <br />
x 3 − 4x 2 +5x − 6<br />
x = −b ± √ b 2 − 4ac<br />
.<br />
2a<br />
<br />
<br />
<br />
2x =6 x =4 <br />
2 · 4 6 ≠8
10/5 =<br />
2 <br />
p q p q <br />
p <br />
q <br />
ax 2 + bx + c =0 a ≠0 <br />
x = −b ± √ b 2 − 4ac<br />
.<br />
2a<br />
ax 2 + bx + c =0 a ≠0 <br />
x = −b ± √ b 2 − 4ac<br />
.<br />
2a<br />
<br />
ax 2 + bx + c =0 a ≠0<br />
<br />
<br />
ax 2 + bx + c =0<br />
x 2 + b a x = − c a<br />
x 2 + b ( ) b 2 ( b 2<br />
a x + = −<br />
2a 2a) c a<br />
(<br />
x + b ) 2<br />
= b2 − 4ac<br />
2a 4a 2<br />
x + b<br />
2a = ±√ b 2 − 4ac<br />
2a<br />
x = −b ± √ b 2 − 4ac<br />
.<br />
2a
s r = s<br />
<br />
p q q <br />
p<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
x x <br />
<br />
A X a A a ∈ A<br />
<br />
x <br />
<br />
X = {x 1 ,x 2 ,...,x n }<br />
x 1 ,x 2 ,...,x n <br />
X = {x : x P}<br />
x X P E <br />
E <br />
E = {2, 4, 6,...} E = {x : x x>0}.
2 ∈ E E −3 /∈ E −3 <br />
E<br />
<br />
N = {n : n } = {1, 2, 3,...};<br />
Z = {n : n } = {...,−1, 0, 1, 2,...};<br />
Q = {r : r } = {p/q : p, q ∈ Z q ≠0};<br />
R = {x : x };<br />
C = {z : z }.<br />
<br />
A B A ⊂ B B ⊃ A A B<br />
<br />
{4, 5, 8} ⊂{2, 3, 4, 5, 6, 7, 8, 9}<br />
<br />
N ⊂ Z ⊂ Q ⊂ R ⊂ C.<br />
B A B ⊂ A <br />
B ≠ A A B A ⊄ B {4, 7, 9} ⊄{2, 4, 5, 8, 9}<br />
A = B A ⊂ B B ⊂ A<br />
<br />
∅ <br />
<br />
A ∪ B A B <br />
A B <br />
A = {1, 3, 5} B = {1, 2, 3, 9} <br />
A ∪ B = {x : x ∈ A x ∈ B};<br />
A ∩ B = {x : x ∈ A x ∈ B}.<br />
A ∪ B = {1, 2, 3, 5, 9} A ∩ B = {1, 3}.<br />
<br />
n⋃<br />
A i = A 1 ∪ ...∪ A n<br />
<br />
i=1<br />
n⋂<br />
A i = A 1 ∩ ...∩ A n<br />
i=1<br />
A 1 ,...,A n <br />
<br />
E O E O <br />
A B A ∩ B = ∅<br />
U <br />
A ⊂ U A A ′ <br />
A ′ = {x : x ∈ U x /∈ A}.<br />
A B <br />
A \ B = A ∩ B ′ = {x : x ∈ A x /∈ B}.
R <br />
<br />
A = {x ∈ R :0
A ∪ B = ∅ A B <br />
(A ∪ B) ′ ⊂ A ′ ∩ B ′ (A ∪ B) ′ ⊃ A ′ ∩ B ′ <br />
x ∈ (A ∪ B) ′ x /∈ A ∪ B x A B <br />
x ∈ A ′ x ∈ B ′ <br />
x ∈ A ′ ∩ B ′ (A ∪ B) ′ ⊂ A ′ ∩ B ′ <br />
x ∈ A ′ ∩ B ′ x ∈ A ′ x ∈ B ′ <br />
x /∈ A x /∈ B x /∈ A ∪ B x ∈ (A ∪ B) ′ (A ∪ B) ′ ⊃ A ′ ∩ B ′ <br />
(A ∪ B) ′ = A ′ ∩ B ′ <br />
<br />
<br />
<br />
<br />
(A \ B) ∩ (B \ A) =∅.<br />
(A \ B) ∩ (B \ A) =(A ∩ B ′ ) ∩ (B ∩ A ′ )<br />
= A ∩ A ′ ∩ B ∩ B ′<br />
= ∅.<br />
A B A × B A<br />
B <br />
A × B = {(a, b) :a ∈ A b ∈ B}.<br />
A = {x, y} B = {1, 2, 3} C = ∅ A × B <br />
{(x, 1), (x, 2), (x, 3), (y, 1), (y, 2), (y, 3)}<br />
<br />
A × C = ∅.<br />
n <br />
A 1 ×···×A n = {(a 1 ,...,a n ):a i ∈ A i i =1,...,n}.<br />
A = A 1 = A 2 = ···= A n A n A ×···×A A <br />
n R 3 <br />
A×B f ⊂ A×B<br />
A B (a, b) ∈ f <br />
a ∈ A b ∈ B <br />
A f B f : A → B A → f B<br />
(a, b) ∈ A × B f(a) =b f : a ↦→ b <br />
A f <br />
f(A) ={f(a) :a ∈ A} ⊂B<br />
f
A<br />
<br />
f<br />
B<br />
a<br />
<br />
b<br />
<br />
c<br />
A<br />
g<br />
B<br />
<br />
a<br />
<br />
b<br />
<br />
c<br />
<br />
A = {1, 2, 3} B = {a, b, c} f<br />
g A B f g 1 ∈ A <br />
B g(1) = a g(1) = b<br />
f : A → B <br />
<br />
f : R → R <br />
f(x) =x 3 f : x ↦→ x 3 <br />
f : Q → Z f(p/q) =p 1/2 = 2/4 <br />
f(1/2) = 1 2 <br />
<br />
<br />
f : A → B f B f(A) =B f <br />
a ∈ A b ∈ B f(a) =b<br />
f a 1 ≠ a 2 f(a 1 ) ≠ f(a 2 )<br />
f(a 1 )=f(a 2 ) a 1 = a 2 <br />
<br />
f : Z → Q f(n) =n/1 f <br />
g : Q → Z g(p/q) =p p/q <br />
g <br />
<br />
f : A → B g : B → C <br />
f g A C (g ◦ f)(x) =g(f(x))
A B C<br />
f<br />
g<br />
<br />
a<br />
X<br />
<br />
<br />
b<br />
c<br />
Y<br />
Z<br />
A<br />
<br />
g ◦ f<br />
C<br />
X<br />
<br />
Y<br />
<br />
Z<br />
<br />
f : A → B g : B → C <br />
g ◦ f : A → C <br />
<br />
f(x) =x 2 g(x) =2x +5 <br />
<br />
(f ◦ g)(x) =f(g(x)) = (2x +5) 2 =4x 2 +20x +25<br />
(g ◦ f)(x) =g(f(x)) = 2x 2 +5.<br />
f ◦ g ≠ g ◦ f<br />
f ◦ g = g ◦ f f(x) =x 3 g(x) = 3√ x<br />
<br />
(f ◦ g)(x) =f(g(x)) = f( 3√ x )=( 3√ x ) 3 = x<br />
<br />
(g ◦ f)(x) =g(f(x)) = g(x 3 )= 3√ x 3 = x.<br />
2 × 2 <br />
( ) a b<br />
A = ,<br />
c d<br />
T A : R 2 → R 2 <br />
T A (x, y) =(ax + by, cx + dy)<br />
(x, y) R 2 <br />
( ) a b x<br />
=<br />
c d)(<br />
y<br />
( ) ax + by<br />
.<br />
cx + dy<br />
R n R m
S = {1, 2, 3} π : S → S <br />
π(1) = 2, π(2) = 1, π(3) = 3.<br />
π <br />
( )<br />
1 2 3<br />
=<br />
π(1) π(2) π(3)<br />
( ) 1 2 3<br />
.<br />
2 1 3<br />
S π : S → S S<br />
f : A → B g : B → C h : C → D <br />
(h ◦ g) ◦ f = h ◦ (g ◦ f)<br />
f g g ◦ f <br />
f g g ◦ f <br />
f g g ◦ f<br />
<br />
<br />
<br />
h ◦ (g ◦ f) =(h ◦ g) ◦ f.<br />
a ∈ A <br />
(h ◦ (g ◦ f))(a) =h((g ◦ f)(a))<br />
= h(g(f(a)))<br />
=(h ◦ g)(f(a))<br />
=((h ◦ g) ◦ f)(a).<br />
f g c ∈ C <br />
a ∈ A (g ◦ f)(a) =g(f(a)) = c g <br />
b ∈ B g(b) =c a ∈ A f(a) =b<br />
<br />
(g ◦ f)(a) =g(f(a)) = g(b) =c.<br />
S id S id S <br />
id(s) =s s ∈ S g : B → A <br />
f : A → B g ◦ f = id A f ◦ g = id B <br />
<br />
f −1 f<br />
f(x) =x 3 f −1 (x) = 3√ x <br />
f(x) = x <br />
f −1 (x) =e x <br />
<br />
f(f −1 (x)) = f(e x )= e x = x<br />
<br />
<br />
f −1 (f(x)) = f −1 ( x) =e x = x
A R 2 R 2 <br />
A =<br />
( ) 3 1<br />
.<br />
5 2<br />
T A (x, y) =(3x + y, 5x +2y).<br />
T A A T −1<br />
A<br />
= T A −1<br />
<br />
( )<br />
A −1 2 −1<br />
=<br />
;<br />
−5 3<br />
<br />
<br />
T −1<br />
A<br />
T −1<br />
A<br />
(x, y) =(2x − y, −5x +3y).<br />
◦ T A(x, y) =T A ◦ T −1 (x, y) =(x, y).<br />
<br />
A<br />
T B (x, y) =(3x, 0)<br />
<br />
( ) 3 0<br />
B = ,<br />
0 0<br />
<br />
T −1<br />
B<br />
(x, y) =(ax + by, cx + dy)<br />
<br />
(x, y) =T ◦ T −1<br />
B<br />
(x, y) =(3ax +3by, 0)<br />
x y y 0<br />
<br />
π =<br />
( 1 2<br />
) 3<br />
2 3 1<br />
S = {1, 2, 3} <br />
( )<br />
π −1 1 2 3<br />
=<br />
3 1 2<br />
π <br />
<br />
<br />
f : A → B g : B → A <br />
g ◦ f = id A g(f(a)) = a a 1 ,a 2 ∈ A f(a 1 )=f(a 2 ) <br />
a 1 = g(f(a 1 )) = g(f(a 2 )) = a 2 f b ∈ B<br />
f a ∈ A f(a) =b f(g(b)) = b<br />
g(b) ∈ A a = g(b)<br />
f b ∈ B f a ∈ A <br />
f(a) =b f a g g(b) =a <br />
f
X <br />
R ⊂ X × X <br />
(x, x) ∈ R x ∈ X <br />
(x, y) ∈ R (y, x) ∈ R <br />
(x, y) (y, z) ∈ R (x, z) ∈ R <br />
R X x ∼ y (x, y) ∈ R<br />
= ≡ ∼ = <br />
<br />
p q r s q s p/q ∼ r/s<br />
ps = qr ∼ <br />
p/q ∼ r/s r/s ∼ t/u q s u ps = qr ru = st<br />
<br />
psu = qru = qst.<br />
s ≠0 pu = qt p/q ∼ t/u<br />
f g R <br />
f(x) ∼ g(x) f ′ (x) =g ′ (x) <br />
∼ f(x) ∼ g(x)<br />
g(x) ∼ h(x) f(x) − g(x) =c 1 g(x) − h(x) =c 2 <br />
c 1 c 2 <br />
f(x) − h(x) =(f(x) − g(x)) + (g(x) − h(x)) = c 1 − c 2<br />
f ′ (x) − h ′ (x) =0 f(x) ∼ h(x)<br />
(x 1 ,y 1 ) (x 2 ,y 2 ) R 2 (x 1 ,y 1 ) ∼ (x 2 ,y 2 ) x 2 1 +y2 1 = x2 2 +y2 2 <br />
∼ R 2 <br />
A B 2 × 2 <br />
2 × 2 A ∼ B <br />
P PAP −1 = B <br />
A =<br />
( 1<br />
) 2<br />
−1 1<br />
( ) −18 33<br />
B =<br />
,<br />
−11 20<br />
A ∼ B PAP −1 = B <br />
P =<br />
( ) 2 5<br />
.<br />
1 3<br />
I 2 × 2 <br />
I =<br />
( ) 1 0<br />
.<br />
0 1<br />
IAI −1 = IAI = A <br />
A ∼ B P PAP −1 = B <br />
A = P −1 BP = P −1 B(P −1 ) −1 .
A ∼ B B ∼ C P Q<br />
PAP −1 = B QBQ −1 = C <br />
C = QBQ −1 = QP AP −1 Q −1 =(QP )A(QP ) −1 ,<br />
<br />
<br />
P X X 1 ,X 2 ,... X i ∩X j =<br />
∅ i ≠ j ⋃ k X k = X ∼ X x ∈ X<br />
[x] ={y ∈ X : y ∼ x} x <br />
<br />
<br />
<br />
∼ X X<br />
X P = {X i } X <br />
X X i <br />
∼ X x ∈ X<br />
x ∈ [x] [x] X = ⋃ x∈X [x]<br />
x, y ∈ X [x] =[y] [x] ∩ [y] =∅ <br />
[x] [y] z ∈ [x] ∩ [y] z ∼ x z ∼ y<br />
x ∼ y [x] ⊂ [y] [y] ⊂ [x] [x] =[y]<br />
<br />
P = {X i } X <br />
x <br />
y y x x ∼ y y ∼ x <br />
x y y z x <br />
z <br />
<br />
<br />
<br />
<br />
(p, q)<br />
(r, s) <br />
<br />
f(x) g(x)<br />
<br />
R 2 (x 1 ,y 1 ) ∼ (x 2 ,y 2 ) x 2 1 + y2 1 =<br />
x 2 2 + y2 2 <br />
<br />
r s n ∈ N r <br />
s n r s n r − s n<br />
r − s = nk k ∈ Z r ≡ s ( n) <br />
41 ≡ 17 ( 8) 41 − 17 = 24 8 <br />
n Z r <br />
r − r =0 n r ≡ s
( n) r − s = −(s − r) n s − r n s ≡ r<br />
( n) r ≡ s ( n) s ≡ t ( n) <br />
k l r − s = kn s − t = ln <br />
r − t n <br />
r − t = r − s + s − t = kn + ln =(k + l)n,<br />
r − t n<br />
3 <br />
[0] = {...,−3, 0, 3, 6,...},<br />
[1] = {...,−2, 1, 4, 7,...},<br />
[2] = {...,−1, 2, 5, 8,...}.<br />
[0] ∪ [1] ∪ [2] = Z [0] [1] [2]<br />
<br />
n <br />
<br />
n <br />
<br />
<br />
<br />
<br />
<br />
A = {x : x ∈ N x },<br />
B = {x : x ∈ N x },<br />
C = {x : x ∈ N x 5}.<br />
<br />
A ∩ B<br />
B ∩ C<br />
A ∪ B<br />
A ∩ (B ∪ C)<br />
A = {a, b, c} B = {1, 2, 3} C = {x} D = ∅ <br />
<br />
A × B<br />
B × A<br />
A × B × C<br />
A × D<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
A B A × B = B × A <br />
A ∪∅= A A ∩∅= ∅<br />
A ∪ B = B ∪ A A ∩ B = B ∩ A<br />
A ∪ (B ∩ C) =(A ∪ B) ∩ (A ∪ C)<br />
A ∩ (B ∪ C) =(A ∩ B) ∪ (A ∩ C)<br />
A ⊂ B A ∩ B = A<br />
(A ∩ B) ′ = A ′ ∪ B ′ <br />
A ∪ B =(A ∩ B) ∪ (A \ B) ∪ (B \ A)
(A ∪ B) × C =(A × C) ∪ (B × C)<br />
(A ∩ B) \ B = ∅<br />
(A ∪ B) \ B = A \ B<br />
A \ (B ∪ C) =(A \ B) ∩ (A \ C)<br />
A ∩ (B \ C) =(A ∩ B) \ (A ∩ C)<br />
(A \ B) ∪ (B \ A) =(A ∪ B) \ (A ∩ B)<br />
f : Q → Q <br />
f <br />
f(p/q) = p +1<br />
p − 2<br />
f(p/q) = p + q<br />
q 2<br />
f(p/q) = 3p<br />
3q<br />
f(p/q) = 3p2<br />
7q 2 − p q<br />
<br />
<br />
f : R → R f(x) =e x<br />
f : Z → Z f(n) =n 2 +3<br />
f : R → R f(x) = x<br />
f : Z → Z f(x) =x 2<br />
f : A → B g : B → C <br />
f −1 g −1 (g ◦ f) −1 = f −1 ◦ g −1 <br />
<br />
f : N → N <br />
f : N → N <br />
R 2 (x 1 ,y 1 ) ∼ (x 2 ,y 2 ) x 2 1 + y2 1 = x2 2 + y2 2<br />
<br />
f : A → B g : B → C <br />
<br />
f g g ◦ f <br />
g ◦ f g <br />
g ◦ f f <br />
g ◦ f f g <br />
<br />
g ◦ f g f <br />
<br />
f(x) = x +1<br />
x − 1 .<br />
f f f ◦ f −1 <br />
f −1 ◦ f<br />
f : X → Y A 1 ,A 2 ⊂ X B 1 ,B 2 ⊂ Y
f(A 1 ∪ A 2 )=f(A 1 ) ∪ f(A 2 )<br />
f(A 1 ∩ A 2 ) ⊂ f(A 1 ) ∩ f(A 2 ) <br />
f −1 (B 1 ∪ B 2 )=f −1 (B 1 ) ∪ f −1 (B 2 ) <br />
f −1 (B) ={x ∈ X : f(x) ∈ B}.<br />
f −1 (B 1 ∩ B 2 )=f −1 (B 1 ) ∩ f −1 (B 2 )<br />
f −1 (Y \ B 1 )=X \ f −1 (B 1 )<br />
<br />
<br />
<br />
x ∼ y R x ≥ y<br />
m ∼ n Z mn > 0<br />
x ∼ y R |x − y| ≤4<br />
m ∼ n Z m ≡ n ( 6)<br />
∼ R 2 (a, b) ∼ (c, d) a 2 +b 2 ≤ c 2 +d 2 <br />
∼ <br />
m × n R n R m <br />
<br />
x ∼ y <br />
y ∼ x x ∼ x<br />
R 2 \{(0, 0)} (x 1 ,y 1 ) ∼ (x 2 ,y 2 )<br />
λ (x 1 ,y 1 )=(λx 2 ,λy 2 ) ∼ <br />
R 2 \(0, 0) <br />
P(R)
2<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
1+2+···+ n =<br />
n(n +1)<br />
2<br />
n n =1<br />
2 3 4 <br />
<br />
n <br />
(n +1) <br />
n =1<br />
1(1 + 1)<br />
1= .<br />
2<br />
n <br />
n(n +1)<br />
1+2+···+ n +(n +1)= + n +1<br />
2<br />
= n2 +3n +2<br />
2<br />
(n +1)[(n +1)+1]<br />
= .<br />
2<br />
(n +1) <br />
<br />
S N <br />
S <br />
<br />
<br />
<br />
S(n) <br />
n ∈ N S(n 0 ) n 0 k<br />
k ≥ n 0 S(k) S(k +1) S(n) n <br />
n 0
n ≥ 3 2 n >n+4 <br />
8=2 3 > 3+4=7,<br />
n 0 =3 2 k >k+4 k ≥ 3 2 k+1 =2· 2 k ><br />
2(k +4) <br />
2(k +4)=2k +8>k+5=(k +1)+4<br />
k n ≥ 3<br />
10 n+1 +3· 10 n +5 9 n ∈ N n =1<br />
10 1+1 +3· 10 + 5 = 135 = 9 · 15<br />
9 10 k+1 +3· 10 k +5 9 k ≥ 1 <br />
9<br />
10 (k+1)+1 +3· 10 k+1 +5=10 k+2 +3· 10 k+1 +50− 45<br />
= 10(10 k+1 +3· 10 k +5)− 45<br />
<br />
(a + b) n =<br />
a b n ∈ N <br />
( n<br />
k)<br />
=<br />
n∑<br />
k=0<br />
( n<br />
k)<br />
a k b n−k ,<br />
n!<br />
k!(n − k)!<br />
<br />
<br />
( ) n +1<br />
=<br />
k<br />
( n<br />
k)<br />
+<br />
( ) n<br />
.<br />
k − 1<br />
<br />
( ( )<br />
n n<br />
+ =<br />
k)<br />
k − 1<br />
n!<br />
k!(n − k)! + n!<br />
(k − 1)!(n − k +1)!<br />
(n +1)!<br />
=<br />
k!(n +1− k)!<br />
( ) n +1<br />
= .<br />
k<br />
n =1 n<br />
1 <br />
(a + b) n+1 =(a + b)(a + b) n<br />
( n∑ ( )<br />
n<br />
=(a + b)<br />
k)a k b n−k<br />
k=0<br />
n∑<br />
( n<br />
n∑<br />
( n<br />
= a<br />
k)<br />
k+1 b n−k + a<br />
k)<br />
k b n+1−k<br />
k=0<br />
k=0
= a n+1 +<br />
= a n+1 +<br />
k=0<br />
n∑<br />
( ) n<br />
a k b n+1−k +<br />
k − 1<br />
n∑<br />
[( n<br />
)<br />
k − 1<br />
k=1<br />
k=1<br />
n+1<br />
∑<br />
( ) n +1<br />
=<br />
a k b n+1−k .<br />
k<br />
n∑<br />
k=1<br />
( n<br />
k)<br />
a k b n+1−k + b n+1<br />
( n<br />
+ a<br />
k)]<br />
k b n+1−k + b n+1<br />
<br />
<br />
S(n) <br />
n ∈ N S(n 0 ) n 0 S(n 0 ),S(n 0 +<br />
1),...,S(k) S(k +1) k ≥ n 0 S(n) <br />
n ≥ n 0 <br />
S Z S <br />
Z <br />
<br />
<br />
<br />
<br />
1 <br />
<br />
S = {n ∈ N : n ≥ 1} 1 ∈ S n ∈ S 0 < 1 <br />
n = n +0
a b b>0 <br />
q r <br />
a = bq + r<br />
0 ≤ r0 a − b · 0 ∈ S<br />
a0.<br />
a − b(q +1) S a − b(q +1)
= a − (ar + bs)q<br />
= a − arq − bsq<br />
= a(1 − rq)+b(−sq),<br />
S d S <br />
r ′ =0 d a d b d <br />
a b<br />
d ′ a b d ′ | d<br />
a = d ′ h b = d ′ k <br />
d = ar + bs = d ′ hr + d ′ ks = d ′ (hr + ks).<br />
d ′ d d a b<br />
a b <br />
r s ar + bs =1<br />
<br />
<br />
<br />
945 2415 <br />
<br />
2415 = 945 · 2 + 525<br />
945 = 525 · 1 + 420<br />
525 = 420 · 1 + 105<br />
420 = 105 · 4+0.<br />
105 420 105 525 105 945 105 2415<br />
105 945 2415 d 945 2415<br />
d 105 (945, 2415) = 105<br />
<br />
r s 945r + 2415s = 105 <br />
105 = 525 + (−1) · 420<br />
= 525 + (−1) · [945 + (−1) · 525]<br />
=2· 525 + (−1) · 945<br />
=2· [2415 + (−2) · 945] + (−1) · 945<br />
=2· 2415 + (−5) · 945.<br />
r = −5 s =2 r s r =41 s = −16 <br />
<br />
(a, b) =d <br />
r 1 >r 2 > ···>r n = d <br />
b = aq 1 + r 1<br />
a = r 1 q 2 + r 2<br />
r 1 = r 2 q 3 + r 3
n−2 = r n−1 q n + r n<br />
r n−1 = r n q n+1 .<br />
r s ar + bs = d <br />
<br />
d = r n<br />
= r n−2 − r n−1 q n<br />
= r n−2 − q n (r n−3 − q n−1 r n−2 )<br />
= −q n r n−3 +(1+q n q n−1 )r n−2<br />
<br />
= ra + sb.<br />
d <br />
a b d a b <br />
<br />
<br />
p p>1 p p <br />
p 1 p n>1 <br />
<br />
a b p p | ab <br />
p | a p | b<br />
p a p | b (a, p) =1<br />
r s ar + ps =1<br />
b = b(ar + ps) =(ab)r + p(bs).<br />
p ab p b =(ab)r + p(bs)<br />
<br />
<br />
p 1 ,p 2 ,...,p n P = p 1 p 2 ···p n +1 P <br />
p i 1 ≤ i ≤ n p i P − p 1 p 2 ···p n =1 <br />
P p ≠ p i<br />
P <br />
n <br />
n>1 <br />
n = p 1 p 2 ···p k ,<br />
p 1 ,...,p k <br />
<br />
n = q 1 q 2 ···q l ,<br />
<br />
k = l q i p i
n <br />
n =2 n <br />
m 1 ≤ m
1 2 +2 2 + ···+ n 2 n(n + 1)(2n +1)<br />
=<br />
6<br />
n ∈ N<br />
<br />
1 3 +2 3 + ···+ n 3 = n2 (n +1) 2<br />
4<br />
n ∈ N<br />
n! > 2 n n ≥ 4<br />
<br />
n(3n − 1)x<br />
x +4x +7x + ···+(3n − 2)x =<br />
2<br />
n ∈ N<br />
10 n+1 +10 n +1 3 n ∈ N<br />
4 · 10 2n +9· 10 2n−1 +5 99 n ∈ N<br />
<br />
n√<br />
a1 a 2 ···a n ≤ 1 n∑<br />
a k .<br />
n<br />
k=1<br />
<br />
<br />
f (n) (x) f (n) n f <br />
(fg) (n) (x) =<br />
n∑<br />
k=0<br />
( n<br />
k)<br />
f (k) (x)g (n−k) (x).<br />
<br />
<br />
1+2+2 2 + ···+2 n =2 n+1 − 1 n ∈ N<br />
<br />
1<br />
2 + 1 6 + ···+ 1<br />
n(n +1) =<br />
n<br />
n +1<br />
n ∈ N<br />
x (1+x) n −1 ≥ nx n =0, 1, 2,...<br />
X X P(X) <br />
X <br />
P({a, b}) ={∅, {a}, {b}, {a, b}}.<br />
n n <br />
2 n <br />
<br />
<br />
<br />
<br />
S ⊂ N <br />
1 ∈ S n +1∈ S n ∈ S S = N<br />
a b (a, b) <br />
r s (a, b) =ra + sb
14 39<br />
234 165<br />
1739 9923<br />
471 562<br />
23771 19945<br />
−4357 3754<br />
a b r s ar + bs =1<br />
a b <br />
<br />
1, 1, 2, 3, 5, 8, 13, 21,....<br />
f 1 =1 f 2 =1 f n+2 = f n+1 + f n n ∈ N<br />
f n < 2 n <br />
f n+1 f n−1 = f 2 n +(−1) n n ≥ 2<br />
f n =[(1+ √ 5) n − (1 − √ 5) n ]/2 n√ 5<br />
n→∞ f n /f n+1 =( √ 5 − 1)/2<br />
f n f n+1 <br />
a b (a, b) =1 r s <br />
ar + bs =1 <br />
(a, s) =(r, b) =(r, s) =1.<br />
x, y ∈ N xy x y <br />
<br />
4k <br />
4k +1 k<br />
a, b, r, s <br />
a 2 + b 2 = r 2<br />
a 2 − b 2 = s 2 .<br />
a r s b <br />
n ∈ N <br />
n 0, 1,...,n− 1 r <br />
s Z 0 ≤ s
p ≥ 2 2 p − 1 p <br />
6n +5<br />
4n − 1<br />
2 p q <br />
p 2 =2q 2 √ 2 <br />
<br />
<br />
<br />
N n 1
3<br />
<br />
<br />
<br />
Z <br />
2 × 2 <br />
2 × 2 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
n<br />
n <br />
<br />
<br />
a b n n <br />
a − b n Z n <br />
Z n 12 <br />
<br />
[0] = {...,−12, 0, 12, 24,...},<br />
[1] = {...,−11, 1, 13, 25,...},<br />
<br />
[11] = {...,−1, 11, 23, 35,...}.<br />
0, 1,...,11 <br />
[0], [1],...,[11] Z n a b <br />
n (a + b) ( n) a + b <br />
n n (ab) ( n) ab <br />
n
n<br />
7+4≡ 1 ( 5) 7 · 3 ≡ 1 ( 5)<br />
3+5≡ 0 ( 8) 3 · 5 ≡ 7 ( 8)<br />
3+4≡ 7 ( 12) 3 · 4 ≡ 0 ( 12)<br />
n<br />
0 n<br />
<br />
Z n <br />
Z 8 2 4 6 <br />
n =2 4 6 k <br />
kn ≡ 1( 8)<br />
· 0 1 2 3 4 5 6 7<br />
0 0 0 0 0 0 0 0 0<br />
1 0 1 2 3 4 5 6 7<br />
2 0 2 4 6 0 2 4 6<br />
3 0 3 6 1 4 7 2 5<br />
4 0 4 0 4 0 4 0 4<br />
5 0 5 2 7 4 1 6 3<br />
6 0 6 4 2 0 6 4 2<br />
7 0 7 6 5 4 3 2 1<br />
Z 8<br />
Z n<br />
a, b, c ∈ Z n <br />
n <br />
<br />
a + b ≡ b + a ( n)<br />
ab ≡ ba ( n).<br />
<br />
(a + b)+c ≡ a +(b + c) ( n)<br />
(ab)c ≡ a(bc) ( n).<br />
<br />
a +0≡ a ( n)<br />
a · 1 ≡ a ( n).<br />
<br />
a(b + c) ≡ ab + ac<br />
( n).
a −a<br />
a +(−a) ≡ 0<br />
( n).<br />
a (a, n) =1 <br />
b a ( n) b <br />
ab ≡ 1<br />
( n).<br />
<br />
<br />
n a + b<br />
n b + a n<br />
(a, n) =1 r s ar + ns =1<br />
ns =1− ar ar ≡ 1( n) b <br />
r ab ≡ 1( n)<br />
b ab ≡ 1( n) n<br />
ab − 1 k ab − nk =1 d = (a, n) d<br />
ab − nk d 1 d =1<br />
<br />
A<br />
B<br />
<br />
A<br />
B<br />
D<br />
C<br />
D<br />
C<br />
A<br />
D<br />
B<br />
C<br />
180 ◦<br />
<br />
C<br />
B<br />
D<br />
A<br />
A<br />
D<br />
B<br />
B<br />
<br />
<br />
C<br />
C<br />
A<br />
D<br />
A<br />
D<br />
B<br />
D<br />
<br />
<br />
C<br />
A<br />
<br />
C<br />
B<br />
<br />
<br />
<br />
180 ◦ 360 ◦
90 ◦ <br />
<br />
A<br />
B<br />
<br />
C A<br />
B<br />
C<br />
id =<br />
( A B<br />
) C<br />
A B C<br />
A<br />
B<br />
<br />
C C<br />
A<br />
B<br />
ρ 1 =<br />
( A B<br />
) C<br />
B C A<br />
A<br />
B<br />
<br />
C B<br />
C<br />
A<br />
ρ 2 =<br />
( A B<br />
) C<br />
C A B<br />
A<br />
B<br />
<br />
C A<br />
C<br />
B<br />
μ 1 =<br />
( A B<br />
) C<br />
A C B<br />
A<br />
B<br />
<br />
C C<br />
B<br />
A<br />
μ 2 =<br />
( A B<br />
) C<br />
C B A<br />
A<br />
B<br />
<br />
C B<br />
A<br />
C<br />
μ 3 =<br />
( A B<br />
) C<br />
B A C<br />
<br />
△ABC <br />
△ABC A B C <br />
<br />
S π : S → S 3! = 6 <br />
<br />
<br />
3·2·1 =3!=6<br />
<br />
A B B C C A <br />
( ) A B C<br />
.<br />
B C A<br />
<br />
120 ◦ <br />
<br />
△ABC <br />
μ 1 ρ 1 <br />
ρ 1 μ 1
(μ 1 ρ 1 )(A) =μ 1 (ρ 1 (A)) = μ 1 (B) =C<br />
(μ 1 ρ 1 )(B) =μ 1 (ρ 1 (B)) = μ 1 (C) =B<br />
(μ 1 ρ 1 )(C) =μ 1 (ρ 1 (C)) = μ 1 (A) =A.<br />
μ 2 <br />
ρ 1 μ 1 μ 3 <br />
ρ 1 μ 1 ≠ μ 1 ρ 1 △ABC <br />
<br />
<br />
α β αβ = <br />
<br />
◦ ρ 1 ρ 2 μ 1 μ 2 μ 3<br />
ρ 1 ρ 2 μ 1 μ 2 μ 3<br />
ρ 1 ρ 1 ρ 2 μ 3 μ 1 μ 2<br />
ρ 2 ρ 2 ρ 1 μ 2 μ 3 μ 1<br />
μ 1 μ 1 μ 2 μ 3 ρ 1 ρ 2<br />
μ 2 μ 2 μ 3 μ 1 ρ 2 ρ 1<br />
μ 3 μ 3 μ 1 μ 2 ρ 1 ρ 2 <br />
<br />
<br />
<br />
n <br />
G G × G → G <br />
(a, b) ∈ G × G a ◦ b ab G <br />
a b (G, ◦) G (a, b) ↦→ a ◦ b <br />
<br />
<br />
a, b, c ∈ G<br />
(a ◦ b) ◦ c = a ◦ (b ◦ c)<br />
e ∈ G <br />
a ∈ G<br />
e ◦ a = a ◦ e = a.<br />
a ∈ G a −1 <br />
<br />
a ◦ a −1 = a −1 ◦ a = e.<br />
G a ◦ b = b ◦ a a, b ∈ G
Z = {...,−1, 0, 1, 2,...} <br />
m, n ∈ Z <br />
+<br />
◦ m+n m◦n 0 <br />
n ∈ Z −n n −1 <br />
m + n = n + m <br />
ab a ◦ b <br />
<br />
m + n −n <br />
m − n m +(−n)<br />
<br />
<br />
n n <br />
Z 5 0 1 2 3 4 <br />
Z 5 <br />
m + n <br />
Z 5 2+3=3+2=0 <br />
Z 5 Z n = {0, 1,...,n− 1} <br />
n<br />
+ 0 1 2 3 4<br />
0 0 1 2 3 4<br />
1 1 2 3 4 0<br />
2 2 3 4 0 1<br />
3 3 4 0 1 2<br />
4 4 0 1 2 3<br />
(Z 5 , +)<br />
<br />
Z n Z n <br />
1 · k = k · 1=k k ∈ Z n <br />
0 0 · k = k · 0=0 k Z n <br />
Z n \{0} 2 ∈ Z 6 <br />
<br />
0 · 2=0 1· 2=2<br />
2 · 2=4 3· 2=0<br />
4 · 2=2 5· 2=4.<br />
k Z n k <br />
n Z n U(n) U(n) <br />
Z n U(8)
· 1 3 5 7<br />
1 1 3 5 7<br />
3 3 1 7 5<br />
5 5 7 1 3<br />
7 7 5 3 1<br />
U(8)<br />
<br />
αβ = βα <br />
α β <br />
<br />
S 3 D 3 <br />
M 2 (R) 2 × 2 GL 2 (R) <br />
M 2 (R) <br />
( ) a b<br />
A =<br />
c d<br />
GL 2 (R) A −1 AA −1 = A −1 A = I I 2 × 2<br />
A <br />
A A = ad − bc ≠0 <br />
<br />
( ) 1 0<br />
I = .<br />
0 1<br />
A ∈ GL 2 (R) <br />
A −1 =<br />
1<br />
ad − bc<br />
( d −b<br />
−c a<br />
<br />
AB = BA<br />
GL 2 (R) <br />
<br />
( ) ( )<br />
1 0<br />
0 1<br />
1=<br />
I =<br />
J =<br />
0 1<br />
( ) 0 i<br />
i 0<br />
K =<br />
)<br />
.<br />
−1 0<br />
( ) i 0<br />
,<br />
0 −i<br />
i 2 = −1 I 2 = J 2 = K 2 = −1 IJ = K JK = I KI = J<br />
JI = −K KJ = −I IK = −J Q 8 = {±1, ±I,±J, ±K} <br />
Q 8 <br />
C ∗ <br />
C ∗ 1 z = a + bi <br />
<br />
z −1 = a − bi<br />
a 2 + b 2<br />
z
G n <br />
|G| = n Z 5 5 Z <br />
|Z| = ∞<br />
<br />
G <br />
e ∈ G eg = ge = g g ∈ G<br />
e e ′ G eg = ge = g e ′ g = ge ′ = g<br />
g ∈ G e = e ′ e ee ′ = e ′ <br />
e ′ ee ′ = e e = ee ′ = e ′ <br />
g ′ g ′′ g<br />
G gg ′ = g ′ g = e gg ′′ = g ′′ g = e g ′ = g ′′ <br />
g ′ = g ′ e = g ′ (gg ′′ )=(g ′ g)g ′′ = eg ′′ = g ′′ <br />
<br />
g G g g −1 <br />
<br />
G a, b ∈ G (ab) −1 = b −1 a −1 <br />
a, b ∈ G abb −1 a −1 = aea −1 = aa −1 = e b −1 a −1 ab = e <br />
(ab) −1 = b −1 a −1 <br />
G a ∈ G (a −1 ) −1 = a<br />
a −1 (a −1 ) −1 = e <br />
a <br />
(a −1 ) −1 = e(a −1 ) −1 = aa −1 (a −1 ) −1 = ae = a.<br />
a b<br />
G x ∈ G ax = b <br />
x <br />
<br />
G a b G <br />
ax = b xa = b G<br />
ax = b x <br />
ax = b a −1 x = ex = a −1 ax = a −1 b<br />
x 1 x 2 ax = b ax 1 =<br />
b = ax 2 x 1 = a −1 ax 1 = a −1 ax 2 = x 2 <br />
xa = b <br />
G a, b, c ∈ G ba = ca b = c ab = ac<br />
b = c
G <br />
g ∈ G g 0 = e n ∈ N <br />
g n = g · g ···g<br />
} {{ }<br />
n <br />
<br />
g −n = g −1 · g −1 ···g −1 .<br />
} {{ }<br />
n <br />
g, h ∈ G<br />
g m g n = g m+n m, n ∈ Z<br />
(g m ) n = g mn m, n ∈ Z<br />
(gh) n =(h −1 g −1 ) −n n ∈ Z G (gh) n = g n h n <br />
(gh) n ≠ g n h n <br />
Z Z n <br />
ng<br />
g n <br />
mg + ng =(m + n)g m, n ∈ Z<br />
m(ng) =(mn)g m, n ∈ Z<br />
m(g + h) =mg + mh n ∈ Z<br />
Z Z n
2Z = {...,−2, 0, 2, 4,...} <br />
<br />
H G H G <br />
G H H G <br />
<br />
H = {e} G <br />
G <br />
<br />
<br />
R ∗ <br />
1 a ∈ R ∗ <br />
1/a <br />
Q ∗ = {p/q : p q }<br />
R ∗ R ∗ 1 1 = 1/1 <br />
R ∗ Q ∗ Q ∗ p/q r/s<br />
pr/qs Q ∗ p/q ∈ Q ∗ Q ∗ <br />
(p/q) −1 = q/p R ∗ Q ∗ <br />
C ∗ <br />
H = {1, −1,i,−i} H C ∗ H <br />
H ⊂ C ∗ <br />
SL 2 (R) GL 2 (R) <br />
<br />
( ) a b<br />
A =<br />
c d<br />
SL 2 (R) ad − bc =1 SL 2 (R) <br />
2×2 <br />
SL 2 (R) A<br />
( )<br />
A −1 d −b<br />
=<br />
.<br />
−c a<br />
<br />
<br />
SL 2 (R) <br />
H G <br />
G H G G <br />
2 × 2 M 2 (R) <br />
2 × 2 M 2 (R) <br />
M 2 (R) <br />
<br />
( 1 0<br />
0 1<br />
)<br />
+<br />
GL 2 (R)<br />
( −1 0<br />
0 −1<br />
)<br />
=<br />
( 0 0<br />
0 0<br />
)<br />
,
Z 4 <br />
0 2 Z 2 <br />
Z 2 × Z 2 <br />
(a, b) +(c, d) =(a + c, b + d) <br />
Z 2 × Z 2 Z 2 × Z 2 <br />
H 1 = {(0, 0), (0, 1)} H 2 = {(0, 0), (1, 0)} H 3 = {(0, 0), (1, 1)} Z 4 Z 2 × Z 2 <br />
<br />
+ (0, 0) (0, 1) (1, 0) (1, 1)<br />
(0, 0) (0, 0) (0, 1) (1, 0) (1, 1)<br />
(0, 1) (0, 1) (0, 0) (1, 1) (1, 0)<br />
(1, 0) (1, 0) (1, 1) (0, 0) (0, 1)<br />
(1, 1) (1, 1) (1, 0) (0, 1) (0, 0)<br />
Z 2 × Z 2<br />
<br />
<br />
H G <br />
<br />
e G H<br />
h 1 ,h 2 ∈ H h 1 h 2 ∈ H<br />
h ∈ H h −1 ∈ H<br />
H G <br />
H e H e H = e <br />
e G e H e H = e H ee H = e H e = e H <br />
ee H = e H e H e = e H <br />
H h ∈ H H <br />
h ′ ∈ H hh ′ = h ′ h = e G<br />
h ′ = h −1 <br />
H <br />
G <br />
<br />
H G H G <br />
H ≠ ∅ g, h ∈ H gh −1 H<br />
H G gh −1 ∈ H<br />
g h H h H h −1 H <br />
gh −1 ∈ H<br />
H ⊂ G H ≠ ∅ gh −1 ∈ H g, h ∈ H <br />
g ∈ H gg −1 = e H g ∈ H eg −1 = g −1 H h 1 ,h 2 ∈ H<br />
H h 1 (h −1<br />
2 )−1 = h 1 h 2 ∈ H <br />
H G
x ∈ Z <br />
3x ≡ 2( 7)<br />
5x +1≡ 13 ( 23)<br />
5x +1≡ 13 ( 26)<br />
9x ≡ 3( 5)<br />
5x ≡ 1( 6)<br />
3x ≡ 1( 6)<br />
G = {a, b, c, d} <br />
<br />
<br />
◦ a b c d<br />
a a c d a<br />
b b b c d<br />
c c d a b<br />
d d a b c<br />
<br />
◦ a b c d<br />
a a b c d<br />
b b c d a<br />
c c d a b<br />
d d a b c<br />
<br />
◦ a b c d<br />
a a b c d<br />
b b a d c<br />
c c d a b<br />
d d c b a<br />
<br />
◦ a b c d<br />
a a b c d<br />
b b a c d<br />
c c b a d<br />
d d d b c<br />
<br />
(Z 4 , +) <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
D 4 <br />
U(12)<br />
S = R \{−1} S a ∗ b = a + b + ab <br />
(S, ∗) <br />
A B GL 2 (R) AB ≠ BA<br />
SL 2 (R)
⎛<br />
1 x<br />
⎞<br />
y<br />
⎝0 1 z⎠<br />
0 0 1<br />
<br />
<br />
⎛ ⎞ ⎛<br />
1 x y 1 x ′ y ′ ⎞ ⎛<br />
1 x + x ′ y + y ′ + xz ′ ⎞<br />
⎝<br />
0 1 z<br />
0 0 1<br />
⎠ ⎝<br />
0 1 z ′<br />
0 0 1<br />
⎠ = ⎝<br />
0 1 z + z ′<br />
0 0 1<br />
(AB) =(A) (B) GL 2 (R) <br />
GL 2 (R) A B GL 2 (R) <br />
AB ∈ GL 2 (R)<br />
Z n 2 = {(a 1,a 2 ,...,a n ):a i ∈ Z 2 } Z n 2 <br />
(a 1 ,a 2 ,...,a n )+(b 1 ,b 2 ,...,b n )=(a 1 + b 1 ,a 2 + b 2 ,...,a n + b n ).<br />
Z n 2 <br />
<br />
R ∗ = R \{0} <br />
R ∗ Z G = R ∗ × Z ◦ G <br />
(a, m) ◦ (b, n) =(ab, m + n) G <br />
<br />
G g, h ∈ G (gh) n ≠ g n h n <br />
<br />
<br />
n! n <br />
<br />
0+a ≡ a +0≡ a ( n)<br />
a ∈ Z n <br />
n<br />
⎠ .<br />
a · 1 ≡ a<br />
( n).<br />
<br />
a ∈ Z n b ∈ Z n <br />
a + b ≡ b + a ≡ 0<br />
( n).<br />
n <br />
<br />
n<br />
n <br />
n<br />
a(b + c) ≡ ab + ac<br />
( n).<br />
<br />
a b G ab n a −1 =(aba −1 ) n n ∈ Z
U(n) Z n n>2 <br />
k ∈ U(n) k 2 =1 k ≠1<br />
g 1 g 2 ···g n gn<br />
−1 gn−1 −1 1 <br />
G a, b ∈ G <br />
xa = b G<br />
<br />
G <br />
G ba = ca b = c ab = ac b = c a, b, c ∈ G<br />
a 2 = e a G G <br />
G a ∈ G a <br />
a 2 = e<br />
G (ab) 2 = a 2 b 2 a b G G<br />
<br />
Z 3 × Z 3 Z 3 × Z 3 <br />
Z 9 <br />
<br />
<br />
H = {2 k : k ∈ Z} H Q ∗ <br />
n =0, 1, 2,... nZ = {nk : k ∈ Z} nZ Z <br />
Z<br />
T = {z ∈ C ∗ : |z| =1} T C ∗ <br />
G 2 × 2 <br />
( )<br />
θ − θ<br />
,<br />
θ θ<br />
θ ∈ R G SL 2 (R)<br />
<br />
G = {a + b √ 2:a, b ∈ Q a b }<br />
R ∗ <br />
G 2 × 2 <br />
{( ) }<br />
a b<br />
H = : a + d =0 .<br />
c d<br />
H G<br />
SL 2 (Z) 2 × 2 <br />
SL 2 (R)<br />
Q 8 <br />
G G<br />
H K G H ∪K <br />
G<br />
H K G HK = {hk : h ∈<br />
H k ∈ K} G G
G g ∈ G <br />
Z(G) ={x ∈ G : gx = xg g ∈ G}<br />
G G<br />
<br />
<br />
<br />
<br />
<br />
a b G a 4 b = ba a 3 = e ab = ba<br />
<br />
xy = x −1 y −1 x y G G <br />
<br />
H G <br />
C(H) ={g ∈ G : gh = hg h ∈ H}.<br />
C(H) G H G<br />
H G g ∈ G gHg −1 = {ghg −1 : h ∈ H} <br />
G<br />
<br />
<br />
<br />
<br />
<br />
d 1 d 2 ···d 12 <br />
<br />
3 · d 1 +1· d 2 +3· d 3 + ···+3· d 11 +1· d 12 ≡ 0 ( 10).<br />
<br />
<br />
<br />
d 12
(d 1 ,d 2 ,...,d k ) · (w 1 ,w 2 ,...,w k ) ≡ 0 ( n)<br />
d 1 w 1 + d 2 w 2 + ···+ d k w k ≡ 0<br />
( n).<br />
(d 1 ,d 2 ,...,d k ) · (w 1 ,w 2 ,...,w k ) ≡ 0( n) <br />
k d 1 d 2 ···d k 0 ≤ d i
4<br />
<br />
Z Z n <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
3 ∈ Z <br />
3 <br />
3Z = {...,−3, 0, 3, 6,...}.<br />
3Z <br />
3 <br />
3 3<br />
H = {2 n : n ∈ Z} H <br />
Q ∗ a =2 m b =2 n H ab −1 =2 m 2 −n =2 m−n<br />
H H Q ∗ 2<br />
G a G <br />
〈a〉 = {a k : k ∈ Z}<br />
G 〈a〉 G a<br />
〈a〉 a 0 = e g h 〈a〉 <br />
〈a〉 g = a m h = a n m n <br />
gh = a m a n = a m+n 〈a〉 g = a n 〈a〉 g −1 = a −n<br />
〈a〉 H G a a<br />
H 〈a〉 〈a〉 G <br />
a<br />
<br />
〈a〉 = {na : n ∈ Z}<br />
a ∈ G 〈a〉 a G <br />
a G = 〈a〉 G a G a<br />
G a n<br />
a n = e |a| = n n <br />
a |a| = ∞ a
1 5 Z 6 Z 6 <br />
2 ∈ Z 6 3 <br />
2 〈2〉 = {0, 2, 4}<br />
Z Z n 1 −1 Z<br />
Z n Z n <br />
Z 6 <br />
U(9) Z 9 U(9) <br />
{1, 2, 4, 5, 7, 8} U(9) <br />
2 1 =2 2 2 =4<br />
2 3 =8 2 4 =7<br />
2 5 =5 2 6 =1.<br />
<br />
S 3 <br />
S 3 <br />
<br />
S 3<br />
{,ρ 1 ,ρ 2 } {,μ 1 } {,μ 2 } {,μ 3 }<br />
{}<br />
S 3<br />
<br />
G a ∈ G G g h G<br />
a g = a r h = a s <br />
G <br />
<br />
gh = a r a s = a r+s = a s+r = a s a r = hg,<br />
<br />
G G G <br />
G <br />
<br />
<br />
G a H <br />
G H = {e} H H <br />
g g a n n H
g −1 = a −n H n −n <br />
H a n>0 m <br />
a m ∈ H m <br />
h = a m H h ′ ∈ H <br />
h h ′ ∈ H H G h ′ = a k <br />
k q r k = mq + r <br />
0 ≤ r
C = {a + bi : a, b ∈ R},<br />
i 2 = −1 z = a + bi a z b <br />
z<br />
z = a + bi w = c + di <br />
<br />
z + w =(a + bi)+(c + di) =(a + c)+(b + d)i.<br />
i 2 = −1 <br />
z w <br />
(a + bi)(c + di) =ac + bdi 2 + adi + bci =(ac − bd)+(ad + bc)i.<br />
z = a + bi <br />
z −1 ∈ C ∗ zz −1 = z −1 z =1z = a + bi <br />
z −1 = a − bi<br />
a 2 + b 2 .<br />
z = a + bi z = a − bi <br />
z = a + bi |z| = √ a 2 + b 2 <br />
z =2+3i w =1− 2i <br />
z + w =(2+3i)+(1− 2i) =3+i<br />
<br />
<br />
zw =(2+3i)(1 − 2i) =8− i.<br />
z −1 = 2<br />
13 − 3 13 i<br />
|z| = √ 13<br />
z =2− 3i.<br />
y<br />
z 3 = −3+2i<br />
z 1 =2+3i<br />
0<br />
x<br />
z 2 =1− 2i
z = a + bi xy a x <br />
b y <br />
z 1 =2+3i z 2 =1− 2i <br />
z 3 = −3+2i <br />
y<br />
r<br />
a + bi<br />
0<br />
θ<br />
x<br />
<br />
<br />
θ x <br />
r <br />
<br />
z = a + bi = r( θ + i θ).<br />
<br />
<br />
r = |z| = √ a 2 + b 2<br />
a = r θ<br />
b = r θ.<br />
r( θ + i θ) r θ z<br />
0 ◦ ≤ θ
z = r θ w = s φ <br />
zw = rs (θ + φ).<br />
z =3(π/3) w =2(π/6) zw =6(π/2) = 6i<br />
z = r θ <br />
n =1, 2,...<br />
[r θ] n = r n (nθ)<br />
n n =1 <br />
k 1 ≤ k ≤ n <br />
z n+1 = z n z<br />
= r n ( nθ + i nθ)r( θ + i θ)<br />
= r n+1 [( nθ θ − nθ θ)+i( nθ θ + nθ θ)]<br />
= r n+1 [(nθ + θ)+i (nθ + θ)]<br />
= r n+1 [(n +1)θ + i (n +1)θ].<br />
z =1+i z 10 <br />
(1 + i) 10 z 10 <br />
<br />
z 10 =(1+i) 10<br />
(√ ( π<br />
)) 10<br />
= 2 <br />
4<br />
=( √ ( ) 5π<br />
2) 10 <br />
2<br />
( π<br />
)<br />
=32<br />
2<br />
=32i.<br />
<br />
C ∗ <br />
Q ∗ R ∗ C ∗ <br />
<br />
T = {z ∈ C : |z| =1}.<br />
<br />
C ∗ <br />
<br />
H = {1, −1,i,−i} H 1 −1 i<br />
−i z 4 =1 <br />
z n =1 n
z n =1 n <br />
( ) 2kπ<br />
z = ,<br />
n<br />
k =0, 1,...,n− 1 n T<br />
n<br />
<br />
<br />
z n = <br />
(<br />
n 2kπ )<br />
= (2kπ) =1.<br />
n<br />
z 2kπ/n <br />
2π z n =1<br />
n n<br />
n T <br />
<br />
n n <br />
<br />
<br />
<br />
√ √<br />
2 2<br />
ω =<br />
2 + 2 i<br />
√ √<br />
2 2<br />
ω 3 = −<br />
2 + 2 i<br />
√ √<br />
2 2<br />
ω 5 = −<br />
2 − 2 i<br />
√ √<br />
2 2<br />
ω 7 =<br />
2 − 2 i.<br />
ω 3<br />
i<br />
y<br />
ω<br />
−1<br />
0 <br />
x<br />
ω 5<br />
−i<br />
ω 7<br />
<br />
<br />
<br />
2 2 <br />
2 8 <br />
2 21,000,000 .
2 37,398,332 ( 46,389),<br />
0 46,388<br />
n <br />
a <br />
2 <br />
a =2 k 1<br />
+2 k 2<br />
+ ···+2 kn ,<br />
k 1
271 28 ≡ 16 ( 481).<br />
271 321 ≡ 271 20 +2 6 +2 8 ( 481)<br />
≡ 271 20 · 271 26 · 271 28 ( 481)<br />
≡ 271 · 419 · 16 ( 481)<br />
≡ 1,816,784 ( 481)<br />
≡ 47 ( 481).<br />
<br />
<br />
n<br />
<br />
<br />
<br />
<br />
<br />
<br />
Z 60 <br />
U(8) <br />
Q <br />
G G <br />
<br />
<br />
<br />
5 ∈ Z 12<br />
√ 3 ∈ R<br />
√ 3 ∈ R ∗<br />
−i ∈ C ∗<br />
72 ∈ Z 240<br />
312 ∈ Z 471<br />
<br />
<br />
Z 7<br />
Z 24 15<br />
Z 12<br />
Z 60<br />
Z 13<br />
Z 48<br />
U(20)
U(18)<br />
R ∗ 7<br />
C ∗ i i 2 = −1<br />
C ∗ 2i<br />
C ∗ (1 + i)/ √ 2<br />
C ∗ (1 + √ 3 i)/2<br />
<br />
<br />
<br />
GL 2 (R) <br />
( 0<br />
) 1<br />
−1 0<br />
( ) 0 1/3<br />
3 0<br />
<br />
<br />
( ) 1 −1<br />
1 0<br />
( ) 1 −1<br />
0 1<br />
<br />
<br />
( 1<br />
) −1<br />
−1 0<br />
(√ )<br />
3/2 1/2<br />
−1/2 √ 3/2<br />
Z 18 <br />
D 4 <br />
Q 8 <br />
U(30)<br />
Z 32 <br />
∗ <br />
<br />
Z Q ∗ R ∗<br />
a 24 = e G a<br />
<br />
n <br />
n ≤ 20 U(n) <br />
<br />
<br />
( )<br />
( )<br />
0 1<br />
0 −1<br />
A =<br />
B =<br />
−1 0<br />
1 −1<br />
GL 2 (R) A B AB <br />
<br />
(3 − 2i)+(5i − 6)<br />
(4 − 5i) − (4i − 4)<br />
(5 − 4i)(7 + 2i)<br />
(9 − i)(9 − i)<br />
i 45<br />
(1 + i)+(1 + i)<br />
<br />
a + bi
2 (π/6)<br />
5 (9π/4)<br />
3 (π)<br />
(7π/4)/2<br />
<br />
<br />
1 − i<br />
−5<br />
2+2i<br />
√ 3+i<br />
−3i<br />
2i +2 √ 3<br />
<br />
<br />
((1 − i)/2) 4<br />
(1 + i) −1<br />
(−i) 10 (−2+2i) −5<br />
(1 − i) 6<br />
( √ 3+i) 5<br />
(− √ 2 − √ 2 i) 12<br />
<br />
<br />
|z| = |z|<br />
zz = |z| 2<br />
z −1 = z/|z| 2<br />
|z + w| ≤|z| + |w|<br />
|z − w| ≥||z|−|w||<br />
|zw| = |z||w|<br />
<br />
<br />
<br />
<br />
<br />
292 3171 ( 582)<br />
2557 341 ( 5681)<br />
2071 9521 ( 4724)<br />
971 321 ( 765)<br />
<br />
a, b ∈ G <br />
a a −1 <br />
g ∈ G |a| = |g −1 ag|<br />
ab ba<br />
p q Z pq <br />
p r Z p r <br />
Z p p <br />
g h 15 16 G <br />
〈g〉∩〈h〉<br />
a G 〈a m 〉∩〈a n 〉<br />
Z n n>2
G a b ∈ G |a| = m |b| = n <br />
(m, n) =1 〈a〉∩〈b〉 = {e}<br />
G G <br />
G<br />
G n x y = x k <br />
(k, n) =1 y G<br />
G 2 <br />
G 4 <br />
G pq (p, q) =1G a<br />
b p q G <br />
Z nZ n =0, 1, 2,...<br />
Z n r 1 ≤ r
5<br />
<br />
<br />
<br />
<br />
△ABC <br />
<br />
S = {A, B, C} π : S → S <br />
<br />
( ) ( ) ( )<br />
A B C A B C A B C<br />
A B C<br />
( A B C<br />
A C B<br />
<br />
C A B B C A<br />
) ( ) A B C<br />
B A C<br />
) ( A B C<br />
C B A<br />
( A B<br />
) C<br />
B C A<br />
A B B C C A <br />
A ↦→ B<br />
B ↦→ C<br />
C ↦→ A.<br />
<br />
<br />
<br />
<br />
X S X X <br />
X = {1, 2,...,n} S n S X <br />
S n n <br />
n S n n! <br />
<br />
S n 1 1 2 2 ... n n <br />
f : S n → S n f −1 f <br />
<br />
|S n | = n!
S n <br />
G S 5 <br />
<br />
( )<br />
1 2 3 4 5<br />
σ =<br />
1 2 3 5 4<br />
( )<br />
1 2 3 4 5<br />
τ =<br />
3 2 1 4 5<br />
( )<br />
1 2 3 4 5<br />
μ =<br />
.<br />
3 2 1 5 4<br />
G<br />
◦ σ τ μ<br />
σ τ μ<br />
σ σ μ τ<br />
τ τ μ σ<br />
μ μ τ σ <br />
<br />
σ τ X <br />
σ τ (σ ◦τ)(x) =σ(τ(x)) τ <br />
σ <br />
στ τ σ <br />
στ(x) σ(τ(x)) <br />
σ(x) (x)σ <br />
<br />
<br />
<br />
( )<br />
1 2 3 4<br />
σ =<br />
4 1 2 3<br />
( )<br />
1 2 3 4<br />
τ =<br />
.<br />
2 1 4 3<br />
<br />
<br />
<br />
στ =<br />
τσ =<br />
( )<br />
1 2 3 4<br />
,<br />
1 4 3 2<br />
( )<br />
1 2 3 4<br />
.<br />
3 2 1 4<br />
<br />
<br />
<br />
σ ∈ S X k a 1 ,a 2 ,...,a k ∈ X<br />
<br />
σ(a 1 )=a 2
σ(a 2 )=a 3<br />
<br />
σ(a k )=a 1<br />
σ(x) =x x ∈ X (a 1 ,a 2 ,...,a k ) <br />
σ <br />
<br />
σ =<br />
6 <br />
( )<br />
1 2 3 4 5 6 7<br />
= (162354)<br />
6 3 5 1 4 2 7<br />
τ =<br />
( )<br />
1 2 3 4 5 6<br />
= (243)<br />
1 4 2 3 5 6<br />
3<br />
<br />
( )<br />
1 2 3 4 5 6<br />
= (1243)(56).<br />
2 4 1 3 6 5<br />
4<br />
<br />
σ = (1352) τ = (256).<br />
σ <br />
τ <br />
1 ↦→ 3, 3 ↦→ 5, 5 ↦→ 2, 2 ↦→ 1,<br />
2 ↦→ 5, 5 ↦→ 6, 6 ↦→ 2,<br />
στ τ σ <br />
1 ↦→ 3, 3 ↦→ 5, 5 ↦→ 6, 6 ↦→ 2 ↦→ 1,<br />
στ = (1356) μ = (1634) σμ = (1652)(34)<br />
S X σ =(a 1 ,a 2 ,...,a k ) τ =(b 1 ,b 2 ,...,b l ) a i ≠ b j <br />
i j<br />
(135) (27) (135) (347)<br />
<br />
(135)(27) = (135)(27)<br />
(135)(347) = (13475).<br />
<br />
<br />
σ τ S X στ = τσ
σ =(a 1 ,a 2 ,...,a k ) τ =(b 1 ,b 2 ,...,b l ) στ(x) =<br />
τσ(x) x ∈ X x {a 1 ,a 2 ,...,a k } {b 1 ,b 2 ,...,b l } σ <br />
τ x σ(x) =x τ(x) =x <br />
στ(x) =σ(τ(x)) = σ(x) =x = τ(x) =τ(σ(x)) = τσ(x).<br />
<br />
x ∈{a 1 ,a 2 ,...,a k }<br />
σ(a i )=a (i k)+1 <br />
a 1 ↦→ a 2<br />
a 2 ↦→ a 3<br />
<br />
a k−1 ↦→ a k<br />
a k ↦→ a 1 .<br />
τ(a i )=a i σ τ <br />
στ(a i )=σ(τ(a i ))<br />
= σ(a i )<br />
= a (i k)+1<br />
= τ(a (i k)+1 )<br />
= τ(σ(a i ))<br />
= τσ(a i ).<br />
x ∈{b 1 ,b 2 ,...,b l } σ τ <br />
S n <br />
X = {1, 2,...,n} σ ∈ S n X 1 <br />
{σ(1),σ 2 (1),...} X 1 X i <br />
X X 1 X 2 {σ(i),σ 2 (i),...} X 2 <br />
X 3 ,X 4 ,... X <br />
<br />
r σ i <br />
{ σ(x) x ∈ Xi<br />
σ i (x) =<br />
x x /∈ X i ,<br />
σ = σ 1 σ 2 ···σ r X 1 ,X 2 ,...,X r σ 1 ,σ 2 ,...,σ r<br />
<br />
<br />
( )<br />
1 2 3 4 5 6<br />
σ =<br />
6 4 3 1 5 2<br />
( )<br />
1 2 3 4 5 6<br />
τ =<br />
.<br />
3 2 1 5 6 4<br />
<br />
σ = (1624)
τ = (13)(456)<br />
στ = (136)(245)<br />
τσ = (143)(256).<br />
<br />
<br />
(1)<br />
<br />
<br />
<br />
(a 1 ,a 2 ,...,a n )=(a 1 a n )(a 1 a n−1 ) ···(a 1 a 3 )(a 1 a 2 ),<br />
<br />
<br />
<br />
<br />
<br />
(16)(253) = (16)(23)(25) = (16)(45)(23)(45)(25).<br />
<br />
(12)(12) (13)(24)(13)(24)<br />
<br />
<br />
(16) <br />
(23)(16)(23)<br />
<br />
(35)(16)(13)(16)(13)(35)(56),<br />
(16) <br />
r <br />
r <br />
= τ 1 τ 2 ···τ r ,<br />
r <br />
r>1 r =2 r>2 <br />
τ r−1 τ r <br />
(ab)(ab) =<br />
(bc)(ab) =(ac)(bc)<br />
(cd)(ab) =(ab)(cd)<br />
(ac)(ab) =(ab)(bc),<br />
a b c d <br />
<br />
τ r−1 τ r <br />
= τ 1 τ 2 ···τ r−3 τ r−2 .
− 2 r <br />
τ r−1 τ r <br />
r <br />
a <br />
τ r−2 τ r−1 r − 2 <br />
r a τ r−2 <br />
r − 2 <br />
τ r−3 τ r−2 <br />
<br />
a <br />
a <br />
r − 2 <br />
<br />
σ <br />
σ <br />
σ <br />
σ <br />
<br />
<br />
<br />
σ = σ 1 σ 2 ···σ m = τ 1 τ 2 ···τ n ,<br />
m n σ σ m ···σ 1 <br />
<br />
= σσ m ···σ 1 = τ 1 ···τ n σ m ···σ 1 ,<br />
n σ <br />
<br />
<br />
<br />
<br />
<br />
S n A n <br />
A n n <br />
A n S n <br />
<br />
A n A n σ <br />
<br />
σ = σ 1 σ 2 ···σ r ,<br />
σ i r <br />
A n <br />
σ −1 = σ r σ r−1 ···σ 1<br />
S n n ≥ 2 <br />
A n n!/2
A n S n B n <br />
<br />
σ S n n ≥ 2 σ <br />
λ σ : A n → B n<br />
<br />
λ σ (τ) =στ.<br />
λ σ (τ) =λ σ (μ) στ = σμ <br />
τ = σ −1 στ = σ −1 σμ = μ.<br />
λ σ λ σ <br />
A 4 S 4 <br />
A 4 <br />
(1) (12)(34) (13)(24) (14)(23)<br />
(123) (132) (124) (142)<br />
(134) (143) (234) (243).<br />
A 4 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
n n n =3, 4,... <br />
n D n <br />
n 1, 2,...,n <br />
n k <br />
k +1 k −1 2n<br />
n
n<br />
1<br />
2<br />
n − 1 3<br />
n<br />
D n S n 2n<br />
4<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
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<br />
<br />
<br />
n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
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<br />
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<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
n<br />
D n n ≥ 3 r <br />
s <br />
r n =1
s 2 =1<br />
srs = r −1 .<br />
n <br />
n <br />
, 360◦<br />
n<br />
, 2 · 360◦<br />
n<br />
360◦<br />
,...,(n − 1) ·<br />
n .<br />
360 ◦ /n r r <br />
<br />
r k = k · 360◦<br />
n .<br />
n s 1 ,s 2 ,...,s n s k k<br />
n <br />
<br />
s 1 = s n/2+1 ,s 2 = s n/2+2 ,...,s n/2 = s n <br />
s 1 ,s 2 ,...,s n <br />
s k s = s 1 s 2 =1 r n =1 <br />
t n k <br />
k +1 k − 1 k +1 t = r k <br />
k − 1 t = sr k r s D n <br />
D n r s<br />
D n = {1,r,r 2 ,...,r n−1 ,s,sr,sr 2 ,...,sr n−1 }.<br />
srs = r −1 <br />
D 4 <br />
1 2 3 4 <br />
r = (1234)<br />
r 2 = (13)(24)<br />
r 3 = (1432)<br />
r 4 =(1)<br />
<br />
s 1 = (24)<br />
s 2 = (13).<br />
D 4 8 <br />
rs 1 = (12)(34)<br />
r 3 s 1 = (14)(23).
D 4<br />
<br />
<br />
n <br />
<br />
6 <br />
<br />
6 · 4=24 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
24 <br />
S 4 <br />
24<br />
S 4 <br />
1 2 3 4 <br />
<br />
S 4 <br />
180 ◦ <br />
<br />
S 4 S 4 <br />
S 4
( 1 2 3 4<br />
) 5<br />
2 4 1 5 3<br />
<br />
( 1 2 3 4<br />
) 5<br />
3 5 1 4 2<br />
<br />
( 1 2 3 4<br />
) 5<br />
4 2 5 1 3<br />
<br />
( 1 2 3 4<br />
) 5<br />
1 4 3 2 5<br />
<br />
<br />
(1345)(234)<br />
(12)(1253)<br />
(143)(23)(24)<br />
(1423)(34)(56)(1324)<br />
(1254)(13)(25)<br />
(1254)(13)(25) 2<br />
(1254) −1 (123)(45)(1254)<br />
(1254) 2 (123)(45)<br />
(123)(45)(1254) −2<br />
(1254) 100<br />
|(1254)|<br />
|(1254) 2 |<br />
(12) −1<br />
(12537) −1<br />
[(12)(34)(12)(47)] −1<br />
[(1235)(467)] −1<br />
<br />
<br />
(14356)<br />
(156)(234)<br />
(1426)(142)<br />
(17254)(1423)(154632)<br />
(142637)
(a 1 ,a 2 ,...,a n ) −1 <br />
S 4 <br />
{σ ∈ S 4 : σ(1) = 3}<br />
{σ ∈ S 4 : σ(2) = 2}<br />
{σ ∈ S 4 : σ(1) = 3 σ(2) = 2}<br />
S 4 <br />
A 4 <br />
S 7 A 7 <br />
A 10 15<br />
A 8 26<br />
S n n =3,...,10<br />
A 5 A 6 <br />
σ ∈ S n n i j σ i = σ j <br />
i ≡ j ( n)<br />
σ = σ 1 ···σ m ∈ S n σ<br />
σ 1 ,...,σ m <br />
D 5 r s <br />
r s<br />
<br />
(12)(34) <br />
<br />
<br />
A 4 <br />
S n n ≥ 3<br />
A n n ≥ 4<br />
D n n ≥ 3<br />
σ ∈ S n σ n − 1<br />
<br />
σ ∈ S n σ σ <br />
n − 2 <br />
σ <br />
σ <br />
σ σ 2 <br />
3 <br />
A n n ≥ 3 3<br />
S n <br />
<br />
(12), (13),...,(1n)<br />
(12), (23),...,(n − 1,n)<br />
(12), (12 ...n)
G λ g : G → G λ g (a) =ga λ g <br />
G<br />
n! n <br />
G <br />
Z(G) ={g ∈ G : gx = xg x ∈ G}.<br />
D 8 D 10 D n <br />
τ =(a 1 ,a 2 ,...,a k ) k<br />
σ <br />
k<br />
στσ −1 =(σ(a 1 ),σ(a 2 ),...,σ(a k ))<br />
μ k σ στσ −1 = μ<br />
α β S n α ∼ β σ ∈ S n σασ −1 = β<br />
∼ S n <br />
σ ∈ S X σ n (x) =y x ∼ y<br />
∼ X<br />
σ ∈ A n τ ∈ S n τ −1 στ ∈ A n <br />
x ∈ X σ ∈ S X <br />
O x,σ = {y : x ∼ y}.<br />
S 5 <br />
α = (1254)<br />
β = (123)(45)<br />
γ = (13)(25).<br />
O x,σ ∩O y,σ ≠ ∅ O x,σ = O y,σ σ <br />
∼<br />
H S X x, y ∈ X σ ∈ H <br />
σ(x) =y 〈σ〉 O x,σ = X x ∈ X<br />
α ∈ S n n ≥ 3 αβ = βα β ∈ S n α <br />
S n <br />
α α −1 α <br />
<br />
α −1 β −1 αβ α, β ∈ S n <br />
r s D n <br />
srs = r −1 <br />
r k s = sr −k D n <br />
r k ∈ D n n/ (k, n)
6<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
G H G H <br />
g ∈ G <br />
gH = {gh : h ∈ H}.<br />
<br />
Hg = {hg : h ∈ H}.<br />
<br />
<br />
H Z 6 0 3 <br />
<br />
0+H =3+H = {0, 3}<br />
1+H =4+H = {1, 4}<br />
2+H =5+H = {2, 5}.<br />
Z Z n <br />
<br />
H S 3 {(1), (123), (132)}<br />
H <br />
(1)H = (123)H = (132)H = {(1), (123), (132)}<br />
(12)H = (13)H = (23)H = {(12), (13), (23)}.<br />
H <br />
H(1) = H(123) = H(132) = {(1), (123), (132)}<br />
H(12) = H(13) = H(23) = {(12), (13), (23)}.
K <br />
S 3 {(1), (12)} K <br />
K <br />
(1)K = (12)K = {(1), (12)}<br />
(13)K = (123)K = {(13), (123)}<br />
(23)K = (132)K = {(23), (132)};<br />
K(1) = K(12) = {(1), (12)}<br />
K(13) = K(132) = {(13), (132)}<br />
K(23) = K(123) = {(23), (123)}.<br />
<br />
<br />
H G g 1 ,g 2 ∈ G <br />
<br />
g 1 H = g 2 H<br />
Hg −1<br />
1 = Hg −1<br />
2 <br />
g 1 H ⊂ g 2 H<br />
g 2 ∈ g 1 H<br />
g −1<br />
1 g 2 ∈ H<br />
H G <br />
<br />
H G H G <br />
G G H G<br />
g 1 H g 2 H H G g 1 H ∩g 2 H =<br />
∅ g 1 H = g 2 H g 1 H ∩ g 2 H ≠ ∅ a ∈ g 1 H ∩ g 2 H <br />
a = g 1 h 1 = g 2 h 2 h 1 h 2 H g 1 = g 2 h 2 h −1<br />
1 <br />
g 1 ∈ g 2 H g 1 H = g 2 H<br />
<br />
G <br />
H<br />
G H G H G <br />
H G [G : H]<br />
G = Z 6 H = {0, 3} [G : H] =3<br />
G = S 3 H = {(1), (123), (132)} K = {(1), (12)} <br />
[G : H] =2 [G : K] =3<br />
H G H G <br />
H G
L H R H H G <br />
φ : L H →R H gH ∈L H <br />
φ(gH) =Hg −1 φ g 1 H = g 2 H <br />
Hg1 −1 = Hg2 −1 φ <br />
Hg −1<br />
1 = φ(g 1 H)=φ(g 2 H)=Hg −1<br />
2 .<br />
g 1 H = g 2 H φ φ(g −1 H)=Hg<br />
<br />
<br />
H G g ∈ G φ : H → gH <br />
φ(h) =gh φ H <br />
gH<br />
φ φ(h 1 )=φ(h 2 ) <br />
h 1 ,h 2 ∈ H h 1 = h 2 φ(h 1 )=gh 1 φ(h 2 )=gh 2 <br />
gh 1 = gh 2 h 1 = h 2 φ <br />
gH gh h ∈ H φ(h) =gh<br />
G H G <br />
|G|/|H| =[G : H] H G <br />
H G<br />
G [G : H] <br />
|H| |G| =[G : H]|H|<br />
G g ∈ G g <br />
G<br />
|G| = p p G g ∈ G <br />
g ≠ e <br />
g G g ≠ e g <br />
|〈g〉| > 1 p g G<br />
p Z p <br />
H K G G ⊃ H ⊃ K<br />
<br />
[G : K] =[G : H][H : K].<br />
<br />
<br />
[G : K] = |G|<br />
|K| = |G|<br />
|H| · |H| =[G : H][H : K].<br />
|K|<br />
A 4 <br />
12 6 <br />
12 1 2 3<br />
4 6 <br />
A 4 6 <br />
H A 4 3 <br />
H 3 H 3 <br />
6
A 4 6<br />
[A 4 : H] =2 H A 4 <br />
H gH = Hg gHg −1 = H<br />
g ∈ A 4 3 A 4 3 H<br />
(123) H (123) −1 = (132) <br />
H ghg −1 ∈ H g ∈ A 4 h ∈ H <br />
(124)(123)(124) −1 = (124)(123)(142) = (243)<br />
(243)(123)(243) −1 = (243)(123)(234) = (142)<br />
H <br />
(1), (123), (132), (243), (243) −1 = (234), (142), (142) −1 = (124).<br />
A 4 6<br />
<br />
τ μ S n <br />
σ ∈ S n μ = στσ −1 <br />
<br />
<br />
τ =(a 1 ,a 2 ,...,a k )<br />
μ =(b 1 ,b 2 ,...,b k ).<br />
σ <br />
σ(a 1 )=b 1<br />
σ(a 2 )=b 2<br />
<br />
σ(a k )=b k .<br />
μ = στσ −1 <br />
τ =(a 1 ,a 2 ,...,a k ) k σ ∈ S n σ(a i )=b <br />
σ(a (i k)+1 )=b ′ μ(b) =b ′ <br />
μ =(σ(a 1 ),σ(a 2 ),...,σ(a k )).<br />
σ μ τ<br />
<br />
<br />
φ φ : N → N φ(n) =1 n =1 n>1<br />
φ(n) m 1 ≤ m
a n n>0 (a, n) =<br />
1 a φ(n) ≡ 1( n)<br />
U(n) φ(n) a φ(n) =1 <br />
a ∈ U(n) a φ(n) − 1 n a φ(n) ≡ 1( n)<br />
n = p <br />
φ(p) =p − 1 <br />
p <br />
p ∤ a p a <br />
a p−1 ≡ 1<br />
( p).<br />
b b p ≡ b ( p)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
G g 5 h <br />
7 |G| ≥35<br />
G 60 <br />
G<br />
<br />
<br />
<br />
〈8〉 Z 24<br />
〈3〉 U(8)<br />
3Z Z<br />
A 4 S 4<br />
A n S n<br />
D 4 S 4<br />
T C ∗<br />
H = {(1), (123), (132)} S 4
SL 2 (R) GL 2 (R) SL 2 (R) GL 2 (R)<br />
n =15 a =4<br />
p =4n +3 <br />
x 2 ≡−1( p)<br />
<br />
<br />
<br />
H G g 1 ,g 2 ∈ G <br />
<br />
g 1 H = g 2 H<br />
Hg1 −1 = Hg2<br />
−1<br />
g 1 H ⊂ g 2 H<br />
g 2 ∈ g 1 H<br />
g −1<br />
1 g 2 ∈ H<br />
ghg −1 ∈ H g ∈ G h ∈ H <br />
gH = Hg g ∈ G<br />
φ : L H →R H φ(gH) =Hg<br />
g n = e g n<br />
α, β ∈ S n <br />
γ β = γαγ −1 β = γαγ −1 γ ∈ S n <br />
α β <br />
|G| =2n 2 <br />
G <br />
[G : H] =2a b H ab ∈ H<br />
[G : H] =2 gH = Hg<br />
H K G gH ∩ gK H ∩ K<br />
G<br />
H K G ∼ G a ∼ b <br />
h ∈ H k ∈ K hak = b <br />
<br />
H = {(1), (123), (132)} A 4 <br />
G n φ(n) <br />
G<br />
n = p e 1<br />
1 pe 2<br />
2 ···pe k<br />
k<br />
p 1 ,p 2 ,...,p k <br />
φ(n) =n<br />
(1 − 1 )(1 − 1 )<br />
···<br />
(1 − 1 )<br />
.<br />
p 1 p 2 p k<br />
<br />
<br />
n = ∑ d|n<br />
φ(d)<br />
n
7<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
... ... <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
A <br />
B C B C A<br />
B<br />
C<br />
<br />
<br />
<br />
A B C <br />
C C <br />
<br />
<br />
<br />
<br />
<br />
f <br />
<br />
f −1 f <br />
f −1 f
=00, =01,..., =<br />
25 <br />
f(p) =p +3 26;<br />
A ↦→ D, B ↦→ E,...,Z ↦→ C <br />
f −1 (p) =p − 3 26 = p +23 26.<br />
<br />
<br />
3, 14, 9, 7, 4, 20, 3.<br />
<br />
0, 11, 6, 4, 1, 17, 0,<br />
3 <br />
26 <br />
<br />
<br />
<br />
<br />
<br />
26 <br />
b f(p) =p + b 26 <br />
=04 <br />
=18 <br />
18 = 4 + b 26 b =14<br />
<br />
<br />
f(p) =p +14 26.<br />
f −1 (p) =p +12 26.<br />
<br />
<br />
<br />
<br />
26 <br />
<br />
<br />
<br />
f(p) =ap + b 26.<br />
f −1 <br />
<br />
c = ap + b 26
p a <br />
(a, 26) = 1 <br />
f −1 (p) =a −1 p − a −1 b 26.<br />
<br />
f(p) =ap + b 26 <br />
a ∈ Z 26 <br />
(a, 26) = 1 a =5 (5, 26) = 1 <br />
a −1 =21 f(p) =5p +3 26<br />
3, 6, 7, 23, 8, 10, 3 <br />
<br />
f −1 (p) =21p − 21 · 3 26 = 21p +15 26.<br />
<br />
<br />
<br />
<br />
p 1 p 2 <br />
=<br />
(<br />
p1<br />
p 2<br />
)<br />
.<br />
A 2 × 2 Z 26 <br />
<br />
f() =A + ,<br />
Z 26 <br />
<br />
f −1 () =A −1 − A −1 .<br />
<br />
7, 4, 11, 15 <br />
( ) 3 5<br />
A = ,<br />
1 2<br />
<br />
( )<br />
A −1 2 21<br />
= .<br />
25 3<br />
=(2, 2)
f <br />
f −1 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
p q n = pq <br />
φ(n) =m =(p − 1)(q − 1) φ φ <br />
E m E <br />
(E,m)=1 D DE ≡ 1<br />
( m) n E <br />
<br />
E n <br />
=00, =02,..., =25 <br />
<br />
n x y = x E n <br />
y x x = y D n <br />
D<br />
<br />
<br />
25 p =23<br />
q =29 <br />
n = pq = 667<br />
<br />
φ(n) =m =(p − 1)(q − 1) = 616.<br />
E = 487 (616, 487) = 1 <br />
25 487 667 = 169.<br />
<br />
191E = 1+151m<br />
(n, D) = (667, 191) <br />
<br />
169 191 667 = 25.<br />
DE ≡ 1( m)<br />
k <br />
DE = km +1=kφ(n)+1.
(x, n) =1 <br />
<br />
y D =(x E ) D = x DE = x km+1 =(x φ(n) ) k x =(1) k x = x n.<br />
x y D n<br />
(x, n) ≠1 n = pq x
2×2 A Z 26 ((A), 26) =<br />
1<br />
<br />
( ) 3 4<br />
A = ,<br />
2 3<br />
f() = A + <br />
=(2, 5) <br />
x x <br />
2 x = 142528 14 25 28 <br />
n = 3551,E = 629,x=31<br />
n = 2257,E =47,x=23<br />
n = 120979,E = 13251,x= 142371<br />
n = 45629,E = 781,x= 231561<br />
<br />
<br />
D <br />
y<br />
n = 3551,D = 1997,y = 2791<br />
n = 5893,D =81,y =34<br />
n = 120979,D = 27331,y = 112135<br />
n = 79403,D = 671,y = 129381<br />
<br />
D<br />
(n, E) <br />
(n, E) = (451, 231)<br />
(n, E) = (3053, 1921)<br />
(n, E) = (37986733, 12371)<br />
(n, E) = (16394854313, 34578451)<br />
n <br />
<br />
n <br />
n E X <br />
X E ≡ X<br />
( n).<br />
<br />
<br />
10 15 (n, E) D
n n n <br />
d =2, 3,..., √ n n d n <br />
n <br />
<br />
n = ab n <br />
<br />
n = x 2 − y 2 =(x − y)(x + y).<br />
<br />
x y n = x 2 − y 2 <br />
<br />
<br />
n <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
p <br />
(a, p) =1 a p−1 ≡ 1( p) <br />
15 <br />
17 <br />
2 15−1 ≡ 2 14 ≡ 4 ( 15).<br />
2 17−1 ≡ 2 16 ≡ 1 ( 17).<br />
n <br />
2 n−1 ≡ 1 ( n).
342<br />
811<br />
<br />
561<br />
771<br />
631<br />
n b (b, n) =1<br />
b n−1 ≡ 1( n) n b 341 <br />
2 3<br />
2000 <br />
2000 <br />
<br />
2000<br />
<br />
<br />
561 = 3 · 11 · 17 <br />
<br />
2163 25 × 10 9
8<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
m <br />
<br />
n <br />
<br />
<br />
<br />
n <br />
<br />
m <br />
<br />
<br />
m n
n <br />
n n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
n (x 1 ,x 2 ,...,x n ) 3n <br />
<br />
(x 1 ,x 2 ,...,x n ) ↦→ (x 1 ,x 2 ,...,x n ,x 1 ,x 2 ,...,x n ,x 1 ,x 2 ,...,x n ).<br />
i i <br />
(0110)<br />
(0110 0110 0110) <br />
(0110 1110 0110) <br />
(0110) <br />
n 2n <br />
<br />
n m <br />
m 3n<br />
<br />
<br />
8 2 8 = 256 8 <br />
2 7 = 128 <br />
<br />
<br />
=65 10 = 01000001 2 ,<br />
=66 10 = 01000010 2 ,<br />
=67 10 = 01000011 2 .<br />
128 <br />
00000000 2 =0 10 ,<br />
<br />
01111111 2 = 127 10 .<br />
0 1<br />
1 <br />
<br />
= 01000001 2 ,<br />
= 01000010 2 ,<br />
n
= 11000011 2 .<br />
<br />
(0100 0101) <br />
1 <br />
<br />
<br />
<br />
m <br />
8 16 32 <br />
0 1 <br />
1 <br />
<br />
1 <br />
<br />
<br />
1<br />
<br />
<br />
0 1 <br />
(1001 1000) <br />
<br />
<br />
<br />
<br />
<br />
000 001 010 011 100 101 110 111<br />
000 0 1 1 2 1 2 2 3<br />
111 3 2 2 1 2 1 1 0<br />
<br />
0 1 0 <br />
(000) 1 (111) <br />
(101) <br />
1 0 <br />
(111) <br />
<br />
(000) (111) <br />
3
p q <br />
pq<br />
n <br />
<br />
<br />
<br />
p<br />
q<br />
q<br />
p<br />
<br />
<br />
<br />
<br />
0 1 p <br />
q =1− p <br />
1 1 p <br />
0 q <br />
n p n p =0.999 <br />
<br />
(0.999) 10,000 ≈ 0.00005.<br />
n (x 1 ,...,x n ) <br />
p <br />
k <br />
( n<br />
q<br />
k)<br />
k p n−k .<br />
k <br />
<br />
k q <br />
n − k p n <br />
q k p n−k k <br />
( n n!<br />
=<br />
k)<br />
k!(n − k)! ,<br />
n k <br />
q k p n−k <br />
( n<br />
q<br />
k)<br />
k p n−k .
p =0.995 500 <br />
<br />
p n =(0.995) 500 ≈ 0.082.<br />
<br />
( n<br />
1)<br />
qp n−1 = 500(0.005)(0.995) 499 ≈ 0.204.<br />
<br />
( n<br />
q<br />
2)<br />
2 p n−2 500 · 499<br />
= (0.005) 2 (0.995) 498 ≈ 0.257.<br />
2<br />
<br />
<br />
1 − 0.082 − 0.204 − 0.257 = 0.457.<br />
<br />
<br />
(n, m) <br />
m n <br />
(n, m) <br />
E : Z m 2 → Z n 2<br />
<br />
D : Z n 2 → Z m 2 .<br />
E E <br />
<br />
D <br />
<br />
(8, 7) <br />
E(x 7 ,x 6 ,...,x 1 )=(x 8 ,x 7 ,...,x 1 ),<br />
x 8 = x 7 + x 6 + ···+ x 1 Z 2 <br />
=(x 1 ,...,x n ) =(y 1 ,...,y n ) n <br />
d(, ) <br />
<br />
d <br />
d(, ) <br />
w() 1 w() =d(, ) <br />
=(00···0)<br />
= (10101) = (11010) = (00011) <br />
C <br />
d(, ) =4, d(, ) =3, d(, ) =3.<br />
<br />
w() =3, w() =3, w() =2.
n <br />
w() =d(, )<br />
d(, ) ≥ 0<br />
d(, ) =0 = <br />
d(, ) =d(, )<br />
d(, ) ≤ d(, )+d(, )<br />
<br />
<br />
<br />
= (1101) = (1100) <br />
(1101) (1100) <br />
(1100) <br />
d(, ) =1 = (1100) <br />
= (1010) d(, ) =2 <br />
<br />
<br />
2 <br />
<br />
0000 0011 0101 0110 1001 1010 1100 1111<br />
0000 0 2 2 2 2 2 2 4<br />
0011 2 0 2 2 2 2 4 2<br />
0101 2 2 0 2 2 4 2 2<br />
0110 2 2 2 0 4 2 2 2<br />
1001 2 2 2 4 0 2 2 2<br />
1010 2 2 4 2 2 0 2 2<br />
1100 2 4 2 2 2 2 0 2<br />
1111 4 2 2 2 2 2 2 0<br />
<br />
<br />
<br />
d(, ) =1 <br />
d(, ) =2 <br />
d =2 <br />
d(, ) =2 <br />
d(, ) =d(, ) =1.<br />
<br />
<br />
d ≥ 3
d(, ) =1 d(, ) ≥ 2 ≠ <br />
<br />
3<br />
C d =2n +1 C n <br />
2n C<br />
n<br />
d(, ) ≤ n <br />
2n +1≤ d(, ) ≤ d(, )+d(, ) ≤ n + d(, ).<br />
d(, ) ≥ n +1 <br />
<br />
2n 1 ≤ d(, ) ≤ 2n 2n+1<br />
1 2n <br />
1 = (00000) 2 = (00111) 3 = (11100)<br />
4 = (11011) <br />
00000 00111 11100 11011<br />
00000 0 3 3 4<br />
00111 3 0 4 3<br />
11100 3 4 0 3<br />
11011 4 3 3 0<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
(32, 6) <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Z n 2 <br />
<br />
n
n <br />
<br />
(11000101) + (11000101) = (00000000).<br />
<br />
(0000000) (0001111) (0010101) (0011010)<br />
(0100110) (0101001) (0110011) (0111100)<br />
(1000011) (1001100) (1010110) (1011001)<br />
(1100101) (1101010) (1110000) (1111111).<br />
Z 7 2<br />
<br />
<br />
d =3 <br />
3 <br />
<br />
<br />
n w( + ) =d(, )<br />
n <br />
<br />
1 <br />
1+1=0<br />
0+0=0<br />
1+0=1<br />
0+1=1.<br />
<br />
d C d <br />
C <br />
d = {w() : ≠ }.<br />
<br />
<br />
d = {d(, ) : ≠ }<br />
= {d(, ) : + ≠ }<br />
= {w( + ) : + ≠ }<br />
= {w() : ≠ }.<br />
<br />
3 <br />
<br />
<br />
<br />
n <br />
· = x 1 y 1 + ···+ x n y n ,
=(x 1 ,x 2 ,...,x n ) =(y 1 ,y 2 ,...,y n ) <br />
= (011001) = (110101) · =0 <br />
<br />
· = <br />
= ( x 1 x 2 ···<br />
⎛ ⎞<br />
y 1<br />
)<br />
y 2<br />
x n ⎜ ⎟<br />
⎝ ⎠<br />
y n<br />
= x 1 y 1 + x 2 y 2 + ···+ x n y n .<br />
3 <br />
3 <br />
1 n =(x 1 ,x 2 ,...,x n ) <br />
1 x 1 + x 2 + ···+ x n =0 4 =(x 1 ,x 2 ,x 3 ,x 4 ) <br />
1 x 1 + x 2 + x 3 + x 4 =0<br />
⎛ ⎞<br />
1<br />
· = = ( )<br />
1<br />
x 1 x 2 x 3 x 4 ⎜ ⎟<br />
⎝1⎠ =0.<br />
1<br />
<br />
<br />
M m×n (Z 2 ) m × n Z 2 <br />
Z 2 <br />
H ∈ M m×n (Z 2 ) n <br />
H = H (H)<br />
<br />
⎛<br />
0 1 0 1<br />
⎞<br />
0<br />
H = ⎝1 1 1 1 0⎠ .<br />
0 0 1 1 1<br />
5 =(x 1 ,x 2 ,x 3 ,x 4 ,x 5 ) H H = <br />
<br />
x 2 + x 4 =0<br />
x 1 + x 2 + x 3 + x 4 =0<br />
x 3 + x 4 + x 5 =0.<br />
5 <br />
(00000) (11110) (10101) (01011).<br />
<br />
H M m×n (Z 2 ) H <br />
n
Z n 2 <br />
, ∈ (H) H M m×n (Z 2 ) H = <br />
H = <br />
H( + ) =H + H = + = .<br />
+ H <br />
H ∈<br />
M m×n (Z 2 )<br />
C <br />
⎛<br />
0 0 0 1 1<br />
⎞<br />
1<br />
H = ⎝0 1 1 0 1 1⎠ .<br />
1 0 1 0 0 1<br />
6 = (010011) <br />
<br />
⎛ ⎞<br />
0<br />
H = ⎝1⎠ ,<br />
1<br />
<br />
<br />
<br />
<br />
<br />
H H <br />
<br />
<br />
H m × n Z 2 n>m m<br />
m × m I m <br />
H =(A | I m ) A m × (n − m) <br />
⎛<br />
⎞<br />
a 11 a 12 ··· a 1,n−m<br />
a 21 a 22 ··· a 2,n−m<br />
⎜<br />
⎝<br />
⎟<br />
⎠<br />
a m1 a m2 ··· a m,n−m<br />
I m m × m <br />
⎛<br />
1 0 ···<br />
⎞<br />
0<br />
0 1 ··· 0<br />
⎜<br />
⎝<br />
⎟<br />
⎠ .<br />
0 0 ··· 1<br />
n × (n − m) <br />
<br />
( )<br />
In−m<br />
G = .<br />
A<br />
G = H = <br />
G
(000), (001), (010),...,(111).<br />
<br />
⎛<br />
0 1<br />
⎞<br />
1<br />
A = ⎝1 1 0⎠ ,<br />
1 0 1<br />
<br />
⎛ ⎞<br />
1 0 0<br />
0 1 0<br />
0 0 1<br />
G =<br />
0 1 1<br />
⎜ ⎟<br />
⎝1 1 0⎠<br />
1 0 1<br />
<br />
⎛<br />
0 1 1 1 0<br />
⎞<br />
0<br />
H = ⎝1 1 0 0 1 0⎠ ,<br />
1 0 1 0 0 1<br />
<br />
H <br />
6 1 1 <br />
=(x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 6 ) <br />
<br />
⎛<br />
⎞<br />
x 2 + x 3 + x 4<br />
= H = ⎝x 1 + x 2 + x 5<br />
⎠ ,<br />
x 1 + x 3 + x 6<br />
x 2 + x 3 + x 4 =0<br />
x 1 + x 2 + x 5 =0<br />
x 1 + x 3 + x 6 =0.<br />
x 4 x 2 x 3 x 5 x 1 x 2 x 6 <br />
x 1 x 3 x 4 x 5 x 6 <br />
x 1 x 2 x 3 x 4 x 5 x 6 <br />
H <br />
(000000) (001101) (010110) (011011)<br />
(100011) (101110) (110101) (111000).<br />
G
G<br />
000 000000<br />
001 001101<br />
010 010110<br />
011 011011<br />
100 100011<br />
101 101110<br />
110 110101<br />
111 111000<br />
<br />
H ∈ M m×n (Z 2 ) (H) <br />
∈ Z n 2 n − m m <br />
H = m <br />
n − m H (n, n − m) <br />
<br />
n − m m <br />
<br />
<br />
G n × k C =<br />
{<br />
: G = ∈ Z<br />
k<br />
2<br />
}<br />
(n, k) C <br />
<br />
G 1 = 1 G 2 = 2 1 + 2 C <br />
G( 1 + 2 )=G 1 + G 2 = 1 + 2 .<br />
<br />
G = G = G = G <br />
G − G = G( − ) =.<br />
k G( − ) x 1 − y 1 ,...,x k − y k <br />
I k G G( − ) = <br />
= <br />
<br />
<br />
( )<br />
H =(A | I m ) m×n G =<br />
In−m<br />
A<br />
n × (n − m) HG = <br />
<br />
C = HG ij C <br />
n∑<br />
c ij = h ik g kj<br />
=<br />
=<br />
k=1<br />
n−m<br />
∑<br />
h ik g kj +<br />
k=1<br />
n−m<br />
∑<br />
k=1<br />
a ik δ kj +<br />
n∑<br />
k=n−m+1<br />
n∑<br />
k=n−m+1<br />
h ik g kj<br />
δ i−(m−n),k a kj
= a ij + a ij<br />
=0,<br />
δ ij =<br />
{<br />
1, i = j<br />
0, i ≠ j<br />
( ) H =(A | I m ) m × n <br />
G =<br />
In−m<br />
A<br />
n × (n − m) H C <br />
G C H = C <br />
H<br />
∈ C G = ∈ Z m 2 <br />
H = HG = <br />
=(y 1 ,...,y n ) H <br />
Z n−m<br />
2 G = H = <br />
<br />
a 11 y 1 + a 12 y 2 + ···+ a 1,n−m y n−m + y n−m+1 =0<br />
a 21 y 1 + a 22 y 2 + ···+ a 2,n−m y n−m + y n−m+1 =0<br />
<br />
a m1 y 1 + a m2 y 2 + ···+ a m,n−m y n−m + y n−m+1 =0.<br />
y n−m+1 ,...,y n y 1 ,...,y n−m <br />
y n−m+1 = a 11 y 1 + a 12 y 2 + ···+ a 1,n−m y n−m<br />
y n−m+1 = a 21 y 1 + a 22 y 2 + ···+ a 2,n−m y n−m<br />
<br />
y n−m+1 = a m1 y 1 + a m2 y 2 + ···+ a m,n−m y n−m .<br />
x i = y i i =1,...,n− m<br />
<br />
H <br />
<br />
1 = (100 ···00) <br />
2 = (010 ···00) <br />
<br />
n = (000 ···01) <br />
n Z n 2 1 m × n H H i i<br />
H<br />
<br />
⎛ ⎞<br />
0<br />
⎛<br />
⎞<br />
1 1 1 0 0<br />
⎜1<br />
⎛ ⎞<br />
⎟ 1<br />
⎝<br />
1 0 0 1 0<br />
1 1 0 0 1<br />
⎠<br />
0<br />
= ⎝0⎠ .<br />
⎜ ⎟<br />
⎝0⎠<br />
1<br />
0
i n 1 i 0<br />
H ∈ M m×n (Z 2 ) H i i H<br />
H m × n H <br />
H <br />
(H) <br />
2 <br />
2 <br />
i i =1,...,n H i i <br />
H i H i <br />
<br />
<br />
H <br />
H i ≠ <br />
<br />
⎛<br />
1 1 1 0<br />
⎞<br />
0<br />
H 1 = ⎝1 0 0 1 0⎠<br />
1 1 0 0 1<br />
<br />
⎛<br />
1 1 1 0<br />
⎞<br />
0<br />
H 2 = ⎝1 0 0 0 0⎠ ,<br />
1 1 0 0 1<br />
H 1 H 2 <br />
<br />
H H 2<br />
3 <br />
<br />
<br />
⎛ ⎞<br />
1 1 1 0<br />
H = ⎝1 0 0 1⎠<br />
1 1 0 0<br />
H <br />
(H) 4<br />
2 (1100) (1010) (1001) (0110) (0101) (0011) <br />
(H) <br />
H H <br />
H <br />
H H <br />
H H <br />
<br />
n i + j 1 i j <br />
w( i + j )=2 i ≠ j <br />
= H( i + j )=H i + H j<br />
i j H
H <br />
<br />
<br />
2 3 =8 <br />
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞<br />
0 1 0 0<br />
⎝ ⎠ , ⎝ ⎠ , ⎝ ⎠ , ⎝ ⎠ .<br />
0<br />
0<br />
0<br />
0<br />
3<br />
H m×n n−m <br />
m 2 m <br />
, 1 ,..., m 2 m − (1 + m)<br />
<br />
<br />
1<br />
0<br />
0<br />
1<br />
<br />
<br />
<br />
n <br />
n <br />
<br />
<br />
<br />
⎛<br />
1 1 1 0<br />
⎞<br />
0<br />
H = ⎝0 1 0 1 0⎠<br />
1 0 0 0 1<br />
5 = (11011) = (01011) <br />
⎛ ⎞<br />
0<br />
⎛ ⎞<br />
1<br />
H = ⎝0⎠ H = ⎝0⎠ .<br />
0<br />
1<br />
<br />
H H <br />
0 1 <br />
H m × n ∈ Z n 2 H <br />
<br />
m × n H <br />
n = + <br />
H <br />
<br />
<br />
<br />
H = H( + ) =H + H = + H = H.
H i <br />
H<br />
H ∈ M m×n (Z 2 ) H<br />
n <br />
<br />
H i i<br />
<br />
<br />
⎛<br />
1 0 1 1 0<br />
⎞<br />
0<br />
H = ⎝0 1 1 0 1 0⎠<br />
1 1 1 0 0 1<br />
6 = (111110) = (111111) = (010111) <br />
<br />
⎛ ⎞ ⎛ ⎞ ⎛ ⎞<br />
1<br />
1<br />
1<br />
H = ⎝1⎠ ,H = ⎝1⎠ ,H = ⎝0⎠ .<br />
1<br />
0<br />
0<br />
<br />
(110110) (010011) <br />
H <br />
<br />
<br />
C <br />
Z n 2 C Zn 2 <br />
C (n, m) C <br />
Z n 2 + C ∈ Zn 2 <br />
2 n−m C Z n 2 <br />
C (5, 3) <br />
<br />
⎛<br />
0 1 1 0<br />
⎞<br />
0<br />
H = ⎝1 0 0 1 0⎠ .<br />
1 1 0 0 1<br />
(00000) (01101) (10011) (11110).<br />
2 5−2 =2 3 C Z 5 2 22 =4
C (00000)(01101)(10011)(11110)<br />
(10000) + C (10000)(11101)(00011)(01110)<br />
(01000) + C (01000)(00101)(11011)(10110)<br />
(00100) + C (00100)(01001)(10111)(11010)<br />
(00010) + C (00010)(01111)(10001)(11100)<br />
(00001) + C (00001)(01100)(10010)(11111)<br />
(10100) + C (00111)(01010)(10100)(11001)<br />
(00110) + C (00110)(01011)(10101)(11000)<br />
C<br />
<br />
n <br />
= + = + <br />
+ C <br />
<br />
n <br />
+ <br />
<br />
<br />
= (01111) (00010) + C<br />
(01101) = (01111) + (00010)<br />
<br />
<br />
<br />
<br />
<br />
<br />
(000) (00000)<br />
(001) (00001)<br />
(010) (00010)<br />
(011) (10000)<br />
(100) (00100)<br />
(101) (01000)<br />
(110) (00110)<br />
(111) (10100)<br />
<br />
C (n, k) H <br />
Z n 2 C H = H <br />
n <br />
n C − ∈ C <br />
H( − ) =0 H = H
C <br />
= (01111) <br />
⎛ ⎞<br />
0<br />
H = ⎝1⎠ .<br />
1<br />
(00010) <br />
<br />
(n, k) <br />
<br />
2 n−k C (32, 24) <br />
2 24 2 32−24 =2 8 = 256 <br />
<br />
<br />
<br />
<br />
<br />
<br />
0 1 2 3 4 5 6 7 8<br />
000 001 010 011 101 110 111 000 001<br />
4 Z 4 2 <br />
<br />
(0110) (1001) (1010) (1100)<br />
<br />
n<br />
(011010), (011100)<br />
(11110101), (01010100)<br />
(00110), (01111)<br />
(1001), (0111)<br />
<br />
n<br />
(011010)<br />
(11110101)<br />
(01111)<br />
(1011)<br />
C 7 <br />
C<br />
<br />
<br />
(011010) (011100) (110111) (110000)<br />
(011100) (011011) (111011) (100011)<br />
(000000) (010101) (110100) (110011)<br />
(000000) (011100) (110101) (110001)
(0110110) (0111100) (1110000) (1111111)<br />
(1001001) (1000011) (0001111) (0000000)<br />
(n, k)<br />
<br />
<br />
<br />
⎛<br />
0 1 0 0<br />
⎞<br />
0<br />
⎝1 0 1 0 1⎠<br />
1 0 0 1 0<br />
<br />
( 1 0 0 1<br />
) 1<br />
0 1 0 1 1<br />
<br />
⎛<br />
⎞<br />
1 0 1 0 0 0<br />
1 1 0 1 0 0<br />
⎜<br />
⎟<br />
⎝0 1 0 0 1 0⎠<br />
1 1 0 0 0 1<br />
<br />
⎛<br />
⎞<br />
0 0 0 1 1 1 1<br />
0 1 1 0 0 1 1<br />
⎜<br />
⎟<br />
⎝1 0 1 0 1 0 1⎠<br />
0 1 1 0 0 1 1<br />
(5, 2) <br />
<br />
C <br />
<br />
⎛<br />
0 1 0 0<br />
⎞<br />
1<br />
H = ⎝1 0 1 0 1⎠ .<br />
0 0 1 1 1<br />
01111 10101 01110 00011<br />
<br />
1000 <br />
p <br />
p =0.01 <br />
p =0.0001<br />
<br />
<br />
<br />
<br />
<br />
<br />
⎛<br />
⎞<br />
1 1 0 0 0<br />
0 0 1 0 0<br />
⎜<br />
⎟<br />
⎝0 0 0 1 0⎠<br />
1 0 0 0 1<br />
⎛<br />
⎞<br />
0 1 1 0 0 0<br />
1 1 0 1 0 0<br />
⎜<br />
⎟<br />
⎝0 1 0 0 1 0⎠<br />
1 1 0 0 0 1<br />
<br />
<br />
( 1 1 1<br />
) 0<br />
1 0 0 1<br />
⎛<br />
⎞<br />
0 0 0 1 0 0 0<br />
0 1 1 0 1 0 0<br />
⎜<br />
⎟<br />
⎝1 0 1 0 0 1 0⎠<br />
0 1 1 0 0 0 1
⎛<br />
⎞<br />
0 1 1 1 1<br />
H = ⎝0 0 0 1 1⎠ .<br />
1 0 1 0 1<br />
<br />
<br />
<br />
<br />
<br />
C Z 3 2 (000) (111) <br />
C Z 3 2 <br />
<br />
C <br />
<br />
<br />
⎛<br />
0 1 0 0<br />
⎞<br />
0<br />
⎝1 0 1 0 1⎠<br />
1 0 0 1 0<br />
<br />
( 1 0 0 1<br />
) 1<br />
0 1 0 1 1<br />
<br />
⎛<br />
⎞<br />
0 0 1 0 0<br />
1 1 0 1 0<br />
⎜<br />
⎟<br />
⎝0 1 0 1 0⎠<br />
1 1 0 0 1<br />
<br />
⎛<br />
⎞<br />
1 0 0 1 1 1 1<br />
1 1 1 0 0 1 1<br />
⎜<br />
⎟<br />
⎝1 0 1 0 1 0 1⎠<br />
1 1 1 0 0 1 0<br />
<br />
n <br />
w() =d(, )<br />
d(, ) =d( + , + )<br />
d(, ) =w( − )<br />
<br />
X d : X × X → R <br />
d(, ) ≥ 0 , ∈ X<br />
d(, ) =0 = <br />
d(, ) =d(, )<br />
d(, ) ≤ d(, )+d(, )
Z n 2 <br />
<br />
C i C<br />
<br />
C <br />
<br />
C <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
i n 1 i 0 <br />
H ∈ M m×n (Z 2 ) H i i H<br />
C (n, k) C <br />
C ⊥ = { ∈ Z n 2 : · =0 ∈ C}.<br />
C C <br />
⎛<br />
1 1 1 0<br />
⎞<br />
0<br />
⎝0 0 1 0 1⎠ .<br />
1 0 0 1 0<br />
C ⊥ (n, n − k) <br />
C C ⊥ <br />
<br />
H m × n Z 2 i i <br />
m <br />
<br />
⎛<br />
0 0 0 1 1<br />
⎞<br />
1<br />
H = ⎝0 1 1 0 0 1⎠<br />
1 0 1 0 1 0<br />
<br />
<br />
<br />
i i <br />
(101011)
H <br />
<br />
(101101) (001001) <br />
(0010000101) (0000101100) <br />
(m, n)<br />
<br />
<br />
k <br />
<br />
<br />
<br />
<br />
<br />
(16, 12)
9<br />
<br />
<br />
Z 4 <br />
T i <br />
<br />
<br />
<br />
<br />
(G, ·) (H, ◦) <br />
φ : G → H <br />
φ(a · b) =φ(a) ◦ φ(b)<br />
a b G G H G ∼ = H φ <br />
<br />
Z 4<br />
∼ = 〈i〉 φ : Z4 →〈i〉 φ(n) =i n <br />
φ φ <br />
<br />
φ(0) = 1<br />
φ(1) = i<br />
φ(2) = −1<br />
φ(3) = −i.<br />
<br />
φ(m + n) =i m+n = i m i n = φ(m)φ(n),<br />
<br />
φ <br />
(R, +) (R + , ·) <br />
<br />
φ(x + y) =e x+y = e x e y = φ(x)φ(y).<br />
φ <br />
<br />
Q ∗ <br />
2 n φ : Z → Q ∗ φ(n) =2 n <br />
φ(m + n) =2 m+n =2 m 2 n = φ(m)φ(n).
φ {2 n : n ∈ Z} Q ∗ <br />
m ≠ n φ(m) ≠ φ(n) <br />
m>n φ(m) =φ(n) 2 m =2 n 2 m−n =1 <br />
m − n>0<br />
Z 8 Z 12 <br />
U(8) ∼ = U(12) <br />
U(8) = {1, 3, 5, 7}<br />
U(12) = {1, 5, 7, 11}.<br />
φ : U(8) → U(12) <br />
1 ↦→ 1<br />
3 ↦→ 5<br />
5 ↦→ 7<br />
7 ↦→ 11.<br />
φ <br />
ψ ψ(1) = 1 ψ(3) = 11 ψ(5) = 5 ψ(7) = 7 <br />
Z 2 × Z 2 <br />
S 3 Z 6 <br />
Z 6 S 3 <br />
φ : Z 6 → S 3 <br />
a, b ∈ S 3 ab ≠ ba φ <br />
m n Z 6 <br />
φ(m) =a φ(n) =b.<br />
<br />
ab = φ(m)φ(n) =φ(m + n) =φ(n + m) =φ(n)φ(m) =ba,<br />
a b <br />
φ : G → H <br />
<br />
<br />
φ −1 : H → G <br />
|G| = |H|<br />
G H <br />
G H <br />
G n H n<br />
φ <br />
<br />
h 1 h 2 H φ <br />
g 1 ,g 2 ∈ G φ(g 1 )=h 1 φ(g 2 )=h 2 <br />
h 1 h 2 = φ(g 1 )φ(g 2 )=φ(g 1 g 2 )=φ(g 2 g 1 )=φ(g 2 )φ(g 1 )=h 2 h 1 .
Z<br />
G a <br />
G φ : Z → G φ : n ↦→ a n <br />
φ(m + n) =a m+n = a m a n = φ(m)φ(n).<br />
φ m n Z m ≠ n<br />
m>n a m ≠ a n <br />
a m = a n a m−n = e m − n>0 <br />
a G a n <br />
n φ(n) =a n <br />
G n G Z n <br />
G n a φ : Z n → G<br />
φ : k ↦→ a k 0 ≤ k < n φ <br />
<br />
G p p G <br />
Z p <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
G <br />
<br />
<br />
G <br />
<br />
Z 3 Z 3 <br />
+ 0 1 2<br />
0 0 1 2<br />
1 1 2 0<br />
2 2 0 1<br />
Z 3 G =<br />
{(0), (012), (021)} <br />
( ) 0 1 2<br />
0 ↦→<br />
=(0)<br />
0 1 2
( ) 0 1 2<br />
1 ↦→<br />
= (012)<br />
1 2 0<br />
( ) 0 1 2<br />
2 ↦→<br />
= (021).<br />
2 0 1<br />
<br />
G G <br />
G g ∈ G λ g : G → G λ g (a) =ga λ g <br />
G λ g λ g (a) =λ g (b) <br />
ga = λ g (a) =λ g (b) =gb.<br />
a = b λ g a ∈ G b <br />
λ g (b) =a b = g −1 a<br />
G <br />
G = {λ g : g ∈ G}.<br />
G <br />
G G <br />
(λ g ◦ λ h )(a) =λ g (ha) =gha = λ gh (a).<br />
<br />
<br />
λ e (a) =ea = a<br />
(λ g −1 ◦ λ g )(a) =λ g −1(ga) =g −1 ga = a = λ e (a).<br />
G G φ : g ↦→ λ g <br />
<br />
φ(gh) =λ gh = λ g λ h = φ(g)φ(h).<br />
φ(g)(a) =φ(h)(a) <br />
ga = λ g a = λ h a = ha.<br />
g = h φ φ(g) =λ g λ g ∈ G<br />
g ↦→ λ g G
G H <br />
G H G × H <br />
<br />
G <br />
G<br />
<br />
(G, ·) (H, ◦) G H <br />
(g, h) ∈ G × H g ∈ G <br />
h ∈ H G × H <br />
(g 1 ,h 1 )(g 2 ,h 2 )=(g 1 · g 2 ,h 1 ◦ h 2 );<br />
G <br />
H · ◦ <br />
(g 1 ,h 1 )(g 2 ,h 2 )=(g 1 g 2 ,h 1 h 2 )<br />
G H G × H <br />
(g 1 ,h 1 )(g 2 ,h 2 )=(g 1 g 2 ,h 1 h 2 ) g 1 ,g 2 ∈ G h 1 ,h 2 ∈ H<br />
e G e H <br />
G H (e G ,e H ) G × H <br />
(g, h) ∈ G × H (g −1 ,h −1 ) <br />
G H<br />
R <br />
R R × R = R 2 <br />
(a, b) +(c, d) =(a + c, b + d) (0, 0) <br />
(a, b) (−a, −b)<br />
<br />
Z 2 × Z 2 = {(0, 0), (0, 1), (1, 0), (1, 1)}.<br />
Z 2 × Z 2 Z 4 <br />
(a, b) Z 2 × Z 2 2 (a, b)+(a, b) =(0, 0) Z 4 <br />
G × H G H <br />
<br />
<br />
n∏<br />
G i = G 1 × G 2 ×···×G n<br />
i=1<br />
G 1 ,G 2 ,...,G n G = G 1 = G 2 =<br />
···= G n G n G 1 × G 2 ×···×G n <br />
Z n 2 n<br />
n <br />
(01011101) + (01001011) = (00010110).
(g, h) ∈ G × H g h r s <br />
(g, h) G × H r s<br />
<br />
<br />
m r s n = |(g, h)|<br />
(g, h) m =(g m ,h m )=(e G ,e H )<br />
(g n ,h n )=(g, h) n =(e G ,e H ).<br />
n m n ≤ m r s <br />
n n r s m <br />
r s m ≤ n m n<br />
(g 1 ,...,g n ) ∈ ∏ G i g i r i G i <br />
(g 1 ,...,g n ) ∏ G i r 1 ,...,r n <br />
(8, 56) ∈ Z 12 × Z 60 (8, 12) = 4 8 12/4 = 3 <br />
Z 12 56 Z 60 15 3 15 15<br />
(8, 56) 15 Z 12 × Z 60 <br />
Z 2 × Z 3 <br />
(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2).<br />
Z 2 × Z 2 Z 4 Z 2 × Z 3<br />
∼ = Z6 <br />
Z 2 × Z 3 (1, 1) Z 2 × Z 3 <br />
<br />
Z m × Z n Z mn (m, n) =1<br />
Z m × Z n<br />
∼ = Zmn (m, n) =1 <br />
(m, n) =d>1 Z m ×Z n <br />
mn/d m n (a, b) ∈ Z m × Z n <br />
(a, b)+(a, b)+···+(a, b) =(0, 0).<br />
} {{ }<br />
mn/d <br />
(a, b) Z m × Z n <br />
(m, n) =mn <br />
(m, n) =1<br />
n 1 ,...,n k <br />
(n i ,n j )=1 i ≠ j<br />
k∏<br />
Z ni<br />
∼ = Zn1···n k<br />
i=1<br />
<br />
m = p e 1<br />
1 ···pe k<br />
k<br />
,<br />
p i <br />
Z m<br />
∼ = Zp e 1<br />
1<br />
×···×Z p<br />
e k<br />
k<br />
.
p e i<br />
i<br />
<br />
p e j<br />
j<br />
i ≠ j <br />
<br />
<br />
Z e p 1 ×···×Z e<br />
1 p k<br />
k<br />
p 1 ,...,p k <br />
<br />
<br />
<br />
<br />
<br />
G H K <br />
G = HK = {hk : h ∈ H, k ∈ K}<br />
H ∩ K = {e}<br />
hk = kh k ∈ K h ∈ H<br />
G H K<br />
U(8) <br />
H = {1, 3} K = {1, 5}.<br />
D 6 <br />
H = {,r 3 } K = {,r 2 ,r 4 ,s,r 2 s, r 4 s}.<br />
K ∼ = S 3 D 6<br />
∼ = Z2 × S 3 <br />
<br />
S 3 <br />
H K H 3 H <br />
{(1), (123), (132)} K 2 <br />
K hk = kh h ∈ H <br />
k ∈ K<br />
G H K G <br />
H × K<br />
G g ∈ G g = hk<br />
h ∈ H k ∈ K φ : G → H × K φ(g) =(h, k)<br />
φ <br />
h k g g = hk = h ′ k ′ <br />
h −1 h ′ = k(k ′ ) −1 H K h = h ′ <br />
k = k ′ φ <br />
φ g 1 = h 1 k 1 g 2 = h 2 k 2 <br />
<br />
φ(g 1 g 2 )=φ(h 1 k 1 h 2 k 2 )<br />
= φ(h 1 h 2 k 1 k 2 )
=(h 1 h 2 ,k 1 k 2 )<br />
=(h 1 ,k 1 )(h 2 ,k 2 )<br />
= φ(g 1 )φ(g 2 ).<br />
φ <br />
Z 6 {0, 2, 4}×{0, 3}<br />
G <br />
H 1 ,H 2 ,...,H n G <br />
G = H 1 H 2 ···H n = {h 1 h 2 ···h n : h i ∈ H i }<br />
H i ∩〈∪ j≠i H j 〉 = {e}<br />
h i h j = h j h i h i ∈ H i h j ∈ H j <br />
<br />
G H i i =1, 2,...,n<br />
G ∏ i H i<br />
<br />
<br />
<br />
<br />
Z ∼ = nZ n ≠0<br />
C ∗ GL 2 (R) <br />
( ) a b<br />
.<br />
−b a<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
U(8) ∼ = Z 4 <br />
U(8) <br />
( ) ( ) ( ) ( )<br />
1 0 1 0 −1 0 −1 0<br />
, , ,<br />
.<br />
0 1 0 −1 0 1 0 −1<br />
U(5) U(10) U(12) <br />
n Z n <br />
n Z n <br />
Q Z<br />
G = R \{−1} G <br />
a ∗ b = a + b + ab.<br />
G (G, ∗) <br />
<br />
<br />
⎛<br />
1 0<br />
⎞<br />
0<br />
⎛<br />
1 0<br />
⎞<br />
0<br />
⎛<br />
0 1<br />
⎞<br />
0<br />
⎝0 1 0⎠<br />
⎝0 0 1⎠<br />
⎝1 0 0⎠<br />
0 0 1 0 1 0 0 0 1
⎛<br />
0 0<br />
⎞<br />
1<br />
⎝1 0 0⎠<br />
0 1 0<br />
⎛<br />
0 0<br />
⎞<br />
1<br />
⎝0 1 0⎠<br />
1 0 0<br />
⎛<br />
0 1<br />
⎞<br />
0<br />
⎝0 0 1⎠<br />
1 0 0<br />
G 6<br />
8<br />
S 4 D 12 <br />
ω = (2π/n) n <br />
A =<br />
( ) ω 0<br />
0 ω −1<br />
B =<br />
D n <br />
<br />
( ) ±1 k<br />
,<br />
0 1<br />
( ) 0 1<br />
1 0<br />
D n Z n <br />
Z 4 × Z 2 <br />
<br />
(3, 4) Z 4 × Z 6<br />
(6, 15, 4) Z 30 × Z 45 × Z 24<br />
(5, 10, 15) Z 25 × Z 25 × Z 25<br />
(8, 8, 8) Z 10 × Z 24 × Z 80<br />
D 4 <br />
Q ∗ 2 m 3 n m, n ∈ Z<br />
Z × Z<br />
S 3 × Z 2 D 6 D 2n <br />
<br />
3 <br />
3<br />
<br />
6<br />
G 20 G H K 4 5<br />
hk = kh h ∈ H k ∈ K G <br />
H K<br />
G H K G × K ∼ =<br />
H × K G ∼ = H<br />
51<br />
52<br />
φ : G → H φ(x) =e H x = e G <br />
e G e H G H <br />
G ∼ = H G H<br />
G p p Z p
S n A n+2 <br />
D n S n <br />
φ : G 1 → G 2 ψ : G 2 → G 3 φ −1 ψ ◦ φ <br />
<br />
<br />
U(5) ∼ = Z 4 U(p) p <br />
S 3 <br />
<br />
G <br />
<br />
φ(a + bi) =a − bi C C<br />
a + ib ↦→ a − ib C ∗ <br />
A ↦→ B −1 AB SL 2 (R) B GL 2 (R)<br />
G (G) (G) <br />
S G G<br />
(Z 6 )<br />
(Z)<br />
G H (G) ∼ = (H)<br />
G g ∈ G i g : G → G i g (x) = gxg −1 <br />
i g G <br />
(G)<br />
(G) (G)<br />
Q 8 (G) =(G)<br />
<br />
G g ∈ G λ g : G → G ρ g : G → G λ g (x) =gx<br />
ρ g (x) =xg −1 i g = ρ g ◦ λ g G <br />
g ↦→ ρ g G<br />
G H K <br />
φ : G → H × K φ(g) =(h, k) g = hk h ∈ H k ∈ K <br />
<br />
G H G n H<br />
n<br />
G ∼ = G H ∼ = H G × H ∼ = G × H<br />
G × H H × G<br />
n 1 ,...,n k <br />
k∏<br />
Z ni<br />
∼ = Zn1···n k<br />
i=1<br />
(n i ,n j )=1 i ≠ j<br />
A × B A B <br />
<br />
∏<br />
G H 1 ,H 2 ,...,H n G <br />
i H i<br />
H 1 H 2 G 1 G 2 H 1 × H 2 <br />
G 1 × G 2
m, n ∈ Z 〈m, n〉 = 〈d〉 d = (m, n)<br />
m, n ∈ Z 〈m〉∩〈n〉 = 〈l〉 l = (m, n)<br />
2p <br />
2p p <br />
G 2p p a ∈ G a<br />
1 2 p 2p<br />
G 2p G Z 2p <br />
G <br />
G 2p G <br />
p G p<br />
G 2p G <br />
2<br />
P G p y ∈ G 2 yP = Py<br />
G 2p P = 〈z〉 <br />
p z y 2 yz = z k y <br />
2 ≤ k
10<br />
<br />
<br />
H G <br />
gH = Hg g ∈ G <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
H G gH = Hg g ∈ G <br />
G <br />
G H G <br />
gh = hg g ∈ G h ∈ H gH = Hg<br />
H S 3 (1) (12) <br />
(123)H = {(123), (13)} H(123) = {(123), (23)},<br />
H S 3 N <br />
(1) (123) (132) N <br />
N = {(1), (123), (132)}<br />
(12)N = N(12) = {(12), (13), (23)}.<br />
<br />
G N G <br />
<br />
N G<br />
g ∈ G gNg −1 ⊂ N<br />
g ∈ G gNg −1 = N
⇒ N G gN = Ng g ∈ G <br />
g ∈ G n ∈ N n ′ N gn = n ′ g gng −1 = n ′ ∈ N<br />
gNg −1 ⊂ N<br />
⇒ g ∈ G gNg −1 ⊂ N N ⊂ gNg −1 n ∈ N<br />
g −1 ng = g −1 n(g −1 ) −1 ∈ N g −1 ng = n ′ n ′ ∈ N n = gn ′ g −1 <br />
gNg −1 <br />
⇒ gNg −1 = N g ∈ G n ∈ N <br />
n ′ ∈ N gng −1 = n ′ gn = n ′ g gN ⊂ Ng <br />
Ng ⊂ gN<br />
<br />
N G N G G/N <br />
(aN)(bN) =abN G<br />
N G/N <br />
N G N G <br />
G/N [G : N]<br />
G/N (aN)(bN) =abN <br />
<br />
aN = bN cN = dN <br />
(aN)(cN) =acN = bdN =(bN)(dN).<br />
a = bn 1 c = dn 2 n 1 n 2 N <br />
acN = bn 1 dn 2 N<br />
= bn 1 dN<br />
= bn 1 Nd<br />
= bNd<br />
= bdN.<br />
eN = N g −1 N <br />
gN G/N N G<br />
<br />
<br />
S 3 N = {(1), (123), (132)} <br />
N S 3 N (12)N S 3 /N <br />
N (12)N<br />
N N (12)N<br />
(12)N (12)N N<br />
Z 2 <br />
S 3 /N <br />
S 3 N = A 3 <br />
(12)N = {(12), (13), (23)} <br />
G/N
3Z Z 3Z Z <br />
0+3Z = {...,−3, 0, 3, 6,...}<br />
1+3Z = {...,−2, 1, 4, 7,...}<br />
2+3Z = {...,−1, 2, 5, 8,...}.<br />
Z/3Z <br />
+ 0+3Z 1+3Z 2+3Z<br />
0+3Z 0+3Z 1+3Z 2+3Z<br />
1+3Z 1+3Z 2+3Z 0+3Z<br />
2+3Z 2+3Z 0+3Z 1+3Z<br />
nZ Z Z/nZ <br />
nZ<br />
1+nZ<br />
2+nZ<br />
<br />
(n − 1) + nZ.<br />
k + Z l + Z k + l + Z <br />
<br />
D n r s<br />
<br />
r n = <br />
s 2 = <br />
srs = r −1 .<br />
r R n D n srs −1 =<br />
srs = r −1 ∈ R n D n D n /R n <br />
Z 2 <br />
<br />
<br />
<br />
Z p <br />
p <br />
<br />
A n <br />
n ≥ 5 <br />
A n 3 n ≥ 3<br />
3 A n <br />
3 (ab) =(ba) <br />
<br />
(ab)(ab) =
(ab)(cd) =(acb)(acd)<br />
(ab)(ac) =(acb).<br />
N A n n ≥ 3 N 3<br />
N = A n <br />
A n 3 (ijk)<br />
i j {1, 2,...,n} k 3 <br />
3 <br />
(iaj) =(ija) 2<br />
(iab) =(ijb)(ija) 2<br />
(jab)=(ijb) 2 (ija)<br />
(abc) =(ija) 2 (ijc)(ijb) 2 (ija).<br />
N A n n ≥ 3 N <br />
3 (ija) N <br />
[(ij)(ak)](ija) 2 [(ij)(ak)] −1 =(ijk)<br />
N N 3 (ijk) 1 ≤ k ≤ n <br />
3 A n N = A n <br />
n ≥ 5 N A n 3<br />
σ N <br />
σ<br />
σ 3<br />
σ σ = τ(a 1 a 2 ···a r ) ∈ N r>3<br />
σ σ = τ(a 1 a 2 a 3 )(a 4 a 5 a 6 )<br />
σ = τ(a 1 a 2 a 3 ) τ <br />
σ = τ(a 1 a 2 )(a 3 a 4 ) τ <br />
σ 3 N σ <br />
σ = τ(a 1 a 2 ···a r ) <br />
N N <br />
N <br />
(a 1 a 2 a 3 )σ(a 1 a 2 a 3 ) −1<br />
σ −1 (a 1 a 2 a 3 )σ(a 1 a 2 a 3 ) −1<br />
σ −1 (a 1 a 2 a 3 )σ(a 1 a 2 a 3 ) −1 = σ −1 (a 1 a 2 a 3 )σ(a 1 a 3 a 2 )<br />
=(a 1 a 2 ···a r ) −1 τ −1 (a 1 a 2 a 3 )τ(a 1 a 2 ···a r )(a 1 a 3 a 2 )<br />
=(a 1 a r a r−1 ···a 2 )(a 1 a 2 a 3 )(a 1 a 2 ···a r )(a 1 a 3 a 2 )<br />
=(a 1 a 3 a r ),
N 3 N = A n <br />
N <br />
σ = τ(a 1 a 2 a 3 )(a 4 a 5 a 6 ).<br />
<br />
<br />
<br />
σ −1 (a 1 a 2 a 4 )σ(a 1 a 2 a 4 ) −1 ∈ N<br />
(a 1 a 2 a 4 )σ(a 1 a 2 a 4 ) −1 ∈ N.<br />
σ −1 (a 1 a 2 a 4 )σ(a 1 a 2 a 4 ) −1 =[τ(a 1 a 2 a 3 )(a 4 a 5 a 6 )] −1 (a 1 a 2 a 4 )τ(a 1 a 2 a 3 )(a 4 a 5 a 6 )(a 1 a 2 a 4 ) −1<br />
=(a 4 a 6 a 5 )(a 1 a 3 a 2 )τ −1 (a 1 a 2 a 4 )τ(a 1 a 2 a 3 )(a 4 a 5 a 6 )(a 1 a 4 a 2 )<br />
=(a 4 a 6 a 5 )(a 1 a 3 a 2 )(a 1 a 2 a 4 )(a 1 a 2 a 3 )(a 4 a 5 a 6 )(a 1 a 4 a 2 )<br />
=(a 1 a 4 a 2 a 6 a 3 ).<br />
N <br />
N σ = τ(a 1 a 2 a 3 ) τ <br />
σ ∈ N σ 2 ∈ N <br />
σ 2 = τ(a 1 a 2 a 3 )τ(a 1 a 2 a 3 )<br />
=(a 1 a 3 a 2 ).<br />
N 3<br />
<br />
σ = τ(a 1 a 2 )(a 3 a 4 ),<br />
τ 2 <br />
σ −1 (a 1 a 2 a 3 )σ(a 1 a 2 a 3 ) −1<br />
N (a 1 a 2 a 3 )σ(a 1 a 2 a 3 ) −1 N <br />
σ −1 (a 1 a 2 a 3 )σ(a 1 a 2 a 3 ) −1 = τ −1 (a 1 a 2 )(a 3 a 4 )(a 1 a 2 a 3 )τ(a 1 a 2 )(a 3 a 4 )(a 1 a 2 a 3 ) −1<br />
=(a 1 a 3 )(a 2 a 4 ).<br />
n ≥ 5 b ∈{1, 2,...,n} b ≠ a 1 ,a 2 ,a 3 ,a 4 μ =(a 1 a 3 b) <br />
<br />
μ −1 (a 1 a 3 )(a 2 a 4 )μ(a 1 a 3 )(a 2 a 4 ) ∈ N<br />
μ −1 (a 1 a 3 )(a 2 a 4 )μ(a 1 a 3 )(a 2 a 4 )=(a 1 ba 3 )(a 1 a 3 )(a 2 a 4 )(a 1 a 3 b)(a 1 a 3 )(a 2 a 4 )<br />
=(a 1 a 3 b).<br />
N 3 <br />
A n n ≥ 5<br />
N A n N 3 <br />
N = A n A n <br />
n ≥ 5
A 5 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
196,833 × 196,833 <br />
<br />
<br />
<br />
<br />
G H G<br />
H G/H<br />
G = S 4 H = A 4<br />
G = A 5 H = {(1), (123), (132)}<br />
G = S 4 H = D 4<br />
G = Q 8 H = {1, −1,I,−I}<br />
G = Z H =5Z<br />
D 4 <br />
D 4 <br />
Q 8 <br />
Q 8 <br />
T 2 × 2 R<br />
<br />
( ) a b<br />
,<br />
0 c<br />
a b c ∈ R ac ≠0 U <br />
( ) 1 x<br />
,<br />
0 1<br />
x ∈ R
U T <br />
U <br />
U T <br />
T /U <br />
T GL 2 (R)<br />
<br />
G G/H <br />
H G H G/H <br />
G <br />
G G/H <br />
H G/H G <br />
H 2 G H <br />
G S n n ≥ 3<br />
G H k H G<br />
g G <br />
C(g) ={x ∈ G : xg = gx}.<br />
C(g) G g G C(g)<br />
G<br />
G <br />
S 3 <br />
GL 2 (R)<br />
Z(G) ={x ∈ G : xg = gx g ∈ G}.<br />
G G<br />
G/Z(G) G <br />
G G ′ = 〈aba −1 b −1 〉 G ′ <br />
G aba −1 b −1 G ′ <br />
G<br />
G ′ G<br />
N G G/N N <br />
G
11<br />
<br />
<br />
<br />
<br />
<br />
<br />
(G, ·) (H, ◦) φ : G → H <br />
φ(g 1 · g 2 )=φ(g 1 ) ◦ φ(g 2 )<br />
g 1 ,g 2 ∈ G φ H φ<br />
<br />
S n<br />
Z 2 S n <br />
Z 2 <br />
<br />
<br />
<br />
<br />
<br />
G g ∈ G φ : Z → G φ(n) =g n <br />
φ <br />
φ(m + n) =g m+n = g m g n = φ(m)φ(n).<br />
Z G g<br />
G = GL 2 (R) <br />
( ) a b<br />
A =<br />
c d<br />
G (A) =ad − bc ≠0 <br />
A B G (AB) =(A) (B) <br />
φ : GL 2 (R) → R ∗ A ↦→ (A)
T z <br />
|z| =1 φ R T<br />
φ : θ ↦→ θ + i θ <br />
φ(α + β) =(α + β)+i (α + β)<br />
=( α β − α β)+i( α β + α β)<br />
=( α + i α)( β + i β)<br />
= φ(α)φ(β).<br />
<br />
<br />
<br />
φ : G 1 → G 2 <br />
e G 1 φ(e) G 2 <br />
g ∈ G 1 φ(g −1 )=[φ(g)] −1 <br />
H 1 G 1 φ(H 1 ) G 2 <br />
H 2 G 2 φ −1 (H 2 )={g ∈ G 1 : φ(g) ∈ H 2 } G 1 <br />
H 2 G 2 φ −1 (H 2 ) G 1 <br />
<br />
e e ′ G 1 G 2 <br />
e ′ φ(e) =φ(e) =φ(ee) =φ(e)φ(e).<br />
φ(e) =e ′ <br />
<br />
φ(g −1 )φ(g) =φ(g −1 g)=φ(e) =e ′ .<br />
φ(H 1 ) G 2 φ(H 1 ) H 1<br />
G 1 x y φ(H 1 ) a, b ∈ H 1 <br />
φ(a) =x φ(b) =y <br />
xy −1 = φ(a)[φ(b)] −1 = φ(ab −1 ) ∈ φ(H 1 ),<br />
φ(H 1 ) G 2 <br />
H 2 G 2 H 1 φ −1 (H 2 ) H 1 <br />
g ∈ G 1 φ(g) ∈ H 2 H 1 φ(e) =e ′ a b H 1 <br />
φ(ab −1 )=φ(a)[φ(b)] −1 H 2 H 2 G 2 ab −1 ∈ H 1<br />
H 1 G 1 H 2 G 2 g −1 hg ∈ H 1 <br />
h ∈ H 1 g ∈ G 1 <br />
φ(g −1 hg) =[φ(g)] −1 φ(h)φ(g) ∈ H 2 ,<br />
H 2 G 2 g −1 hg ∈ H 1 <br />
φ : G → H e H <br />
φ −1 ({e}) G φ <br />
φ G <br />
H
φ : G → H φ <br />
G<br />
φ : GL 2 (R) → R ∗ A ↦→<br />
(A) 1 R ∗ 2 × 2 <br />
φ = SL 2 (R)<br />
φ : R → C ∗ φ(θ) =<br />
θ + i θ {2πn : n ∈ Z} φ ∼ = Z<br />
φ <br />
Z 7 Z 12 φ Z 7 <br />
{0} Z 7 Z 7 Z 12 <br />
Z 12 <br />
7 Z 7 Z 12 <br />
<br />
G g ∈ G φ Z<br />
G φ(n) =g n g <br />
{0} φ Z G g <br />
g n φ nZ<br />
<br />
<br />
<br />
<br />
φ : G → H G φ <br />
G <br />
<br />
H G <br />
φ : G → G/H<br />
<br />
φ(g) =gH.<br />
<br />
φ(g 1 g 2 )=g 1 g 2 H = g 1 Hg 2 H = φ(g 1 )φ(g 2 ).<br />
H <br />
<br />
ψ : G → H <br />
K = ψ K G φ : G → G/K <br />
η : G/K → ψ(G) ψ = ηφ<br />
K G η : G/K → ψ(G) η(gK) =<br />
ψ(g) η g 1 K = g 2 K k ∈ K<br />
g 1 k = g 2 <br />
η(g 1 K)=ψ(g 1 )=ψ(g 1 )ψ(k) =ψ(g 1 k)=ψ(g 2 )=η(g 2 K).
η η : G/K → ψ(G)<br />
ψ = ηφ η <br />
η(g 1 Kg 2 K)=η(g 1 g 2 K)<br />
= ψ(g 1 g 2 )<br />
= ψ(g 1 )ψ(g 2 )<br />
= η(g 1 K)η(g 2 K).<br />
η ψ(G) η η(g 1 K)=η(g 2 K)<br />
ψ(g 1 )=ψ(g 2 ) ψ(g1 −1 g 2)=e g1 −1 g 2 ψ <br />
g1 −1 g 2K = K g 1 K = g 2 K<br />
<br />
ψ = ηφ<br />
G<br />
ψ<br />
H<br />
φ<br />
η<br />
G/K<br />
G g φ : Z → G <br />
n ↦→ g n <br />
φ(m + n) =g m+n = g m g n = φ(m)φ(n).<br />
φ |g| = m g m = e φ = mZ Z/ φ = Z/mZ ∼ = G<br />
g φ =0 φ <br />
G Z <br />
Z Z n <br />
H G<br />
G N G HN <br />
G H ∩ N H <br />
H/H ∩ N ∼ = HN/N.<br />
HN = {hn : h ∈ H, n ∈ N} G <br />
h 1 n 1 ,h 2 n 2 ∈ HN N (h 2 ) −1 n 1 h 2 ∈ N <br />
(h 1 n 1 )(h 2 n 2 )=h 1 h 2 ((h 2 ) −1 n 1 h 2 )n 2<br />
HN hn ∈ HN HN <br />
(hn) −1 = n −1 h −1 = h −1 (hn −1 h −1 ).
H ∩ N H h ∈ H n ∈ H ∩ N <br />
h −1 nh ∈ H H h −1 nh ∈ N N G <br />
h −1 nh ∈ H ∩ N<br />
φ H HN/N h ↦→ hN φ <br />
hnN = hN h H φ <br />
φ(hh ′ )=hh ′ N = hNh ′ N = φ(h)φ(h ′ ).<br />
φ H/ φ <br />
HN/N = φ(H) ∼ = H/ φ.<br />
<br />
φ = {h ∈ H : h ∈ N} = H ∩ N,<br />
HN/N = φ(H) ∼ = H/H ∩ N<br />
N G<br />
H ↦→ H/N H <br />
N G/N G N<br />
G/N<br />
H G N N H H/N <br />
aN bN H/N (aN)(b −1 N)=ab −1 N ∈ H/N <br />
H/N G/N<br />
S G/N N H = {g ∈ G :<br />
gN ∈ S} h 1 ,h 2 ∈ H (h 1 N)(h 2 N)=h 1 h 2 N ∈ S h −1<br />
1 N ∈ S<br />
H G H N S = H/N<br />
H ↦→ H/N <br />
H 1 H 2 G N H 1 /N = H 2 /N<br />
h 1 ∈ H 1 h 1 N ∈ H 1 /N h 1 N = h 2 N ⊂ H 2 h 2 H 2 <br />
N H 2 h 1 ∈ H 2 H 1 ⊂ H 2 H 2 ⊂ H 1 <br />
H 1 = H 2 H ↦→ H/N <br />
H G N H <br />
G/N → G/H gN ↦→ gH <br />
H/N H/N G/N<br />
H/N G/N <br />
G → G/N → G/N<br />
H/N<br />
H H G<br />
<br />
<br />
G N H <br />
G N ⊂ H <br />
G/H ∼ = G/N<br />
H/N .<br />
<br />
Z/mZ ∼ = (Z/mnZ)/(mZ/mnZ).<br />
|Z/mnZ| = mn |Z/mZ| = m |mZ/mnZ| = n
(AB) = (A) (B) A, B ∈ GL 2 (R) <br />
GL 2 (R) R ∗ <br />
<br />
<br />
φ : R ∗ → GL 2 (R) <br />
φ : R → GL 2 (R) <br />
φ : GL 2 (R) → R <br />
φ : GL 2 (R) → R ∗ <br />
φ(a) =<br />
φ(a) =<br />
( ) 1 0<br />
0 a<br />
( ) 1 0<br />
a 1<br />
(( )) a b<br />
φ<br />
= a + d<br />
c d<br />
(( )) a b<br />
φ<br />
= ad − bc<br />
c d<br />
φ : M 2 (R) → R <br />
(( )) a b<br />
φ<br />
= b,<br />
c d<br />
M 2 (R) 2 × 2 R<br />
A m × n x ↦→ Ax <br />
φ : R n → R m <br />
φ : Z → Z φ(n) =7n φ <br />
φ<br />
Z 24 Z 18 <br />
Z Z 12 <br />
Z 24 H = 〈4〉 N = 〈6〉<br />
HN H + N <br />
H ∩ N<br />
HN/N <br />
H/(H ∩ N)
HN/N H/(H ∩ N) <br />
<br />
G n ∈ N φ : G → G g ↦→ g n <br />
<br />
φ : G → H G φ(G) <br />
<br />
φ : G → H G φ(G) <br />
<br />
<br />
<br />
G H k H G<br />
Q/Z ∼ = Q<br />
G N G H G/N<br />
φ −1 (H) G |H|·|N| φ : G → G/N <br />
<br />
G 1 G 2 H 1 H 2 G 1 <br />
G 2 φ : G 1 → G 2 φ <br />
φ :(G 1 /H 1 ) → (G 2 /H 2 ) φ(H 1 ) ⊂ H 2 <br />
H K G H ∩ K = {e} G <br />
G/H × G/K<br />
φ : G 1 → G 2 H 1 <br />
G 1 φ(H 1 )=H 2 G 1 /H 1<br />
∼ = G2 /H 2 <br />
φ : G → H φ <br />
φ −1 (e) ={e}<br />
φ : G → H ∼ G a ∼ b φ(a) =φ(b)<br />
a, b ∈ G <br />
<br />
<br />
<br />
(G) G G <br />
G<br />
(G) ≤ S G <br />
G<br />
i g : G → G,<br />
<br />
i g (x) =gxg −1 ,<br />
g ∈ G i g ∈ (G)<br />
(G) (G) <br />
(G)<br />
G <br />
G i g G <br />
G → (G)
g ↦→ i g .<br />
(G) Z(G) <br />
<br />
G/Z(G) ∼ = (G).<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
(S 3 ) (S 3 ) D 4 <br />
φ : Z → Z (Z)<br />
Z 8 (Z 8 ) ∼ = U(8)<br />
k ∈ Z n φ k : Z n → Z n a ↦→ ka φ k <br />
φ k k Z n <br />
Z n φ k k Z n <br />
ψ : U(n) → (Z n ) ψ : k ↦→ φ k
12<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
T : R n → R m <br />
R n α ∈ R<br />
T ( + ) =T ()+T ()<br />
T (α) =αT ().<br />
m × n R R n R m <br />
=(x 1 ,...,x n ) =(y 1 ,...,y n ) R n <br />
m × n <br />
⎛<br />
⎞<br />
a 11 a 12 ··· a 1n<br />
a 21 a 22 ··· a 2n<br />
A = ⎜<br />
⎝<br />
⎟<br />
⎠<br />
a m1 a m2 ··· a mn<br />
R m α <br />
A( + ) =A + A αA = A(α),<br />
<br />
⎛ ⎞<br />
x 1<br />
x 2<br />
= ⎜ ⎟<br />
⎝ ⎠ .<br />
x n
A (a ij )<br />
T : R n → R m A T <br />
T <br />
1 =(1, 0,...,0) <br />
2 =(0, 1,...,0) <br />
<br />
n =(0, 0,...,1) .<br />
=(x 1 ,...,x n ) <br />
<br />
<br />
x 1 1 + x 2 2 + ···+ x n n .<br />
T ( 1 )=(a 11 ,a 21 ,...,a m1 ) ,<br />
T ( 2 )=(a 12 ,a 22 ,...,a m2 ) ,<br />
<br />
T ( n )=(a 1n ,a 2n ,...,a mn ) ,<br />
T () =T (x 1 1 + x 2 2 + ···+ x n n )<br />
= x 1 T ( 1 )+x 2 T ( 2 )+···+ x n T ( n )<br />
( n∑<br />
) <br />
n∑<br />
= a 1k x k ,..., a mk x k<br />
k=1<br />
k=1<br />
= A.<br />
T : R 2 → R 2 <br />
T (x 1 ,x 2 )=(2x 1 +5x 2 , −4x 1 +3x 2 ),<br />
T <br />
T 1 =(2, −4) T 2 =(5, 3) T <br />
A =<br />
( ) 2 5<br />
.<br />
−4 3<br />
<br />
n × n A <br />
A −1 AA −1 = A −1 A = I <br />
⎛<br />
⎞<br />
1 0 ··· 0<br />
0 1 ··· 0<br />
I = ⎜<br />
⎝<br />
⎟<br />
⎠<br />
0 0 ··· 1<br />
n × n A <br />
A
A <br />
A <br />
A −1 =<br />
( ) 2 1<br />
,<br />
5 3<br />
( ) 3 −1<br />
.<br />
−5 2<br />
A −1 (A) =2· 3 − 5 · 1=1 <br />
<br />
A B n × n <br />
<br />
<br />
(AB) =( A)( B)<br />
A (A −1 )=1/ A<br />
A =(a ij ) A =(a ji ) (A )= A<br />
T n × n A T<br />
| A| R 2 T<br />
| A|<br />
<br />
<br />
2 × 2 <br />
<br />
<br />
n × n <br />
GL n (R) <br />
<br />
<br />
(A) =1 (B) =1 (AB) =(A) (B) =1 (A −1 )=<br />
1/ A =1 SL n (R)<br />
2 × 2 <br />
( ) a b<br />
A = ,<br />
c d<br />
A ad − bc GL 2 (R) <br />
ad − bc ≠0 A <br />
A −1 =<br />
1<br />
ad − bc<br />
( d −b<br />
−c a<br />
)<br />
.<br />
A SL 2 (R) <br />
( )<br />
A −1 d −b<br />
=<br />
.<br />
−c a
SL 2 (R) <br />
A =<br />
( ) 1 1<br />
0 1<br />
SL 2 (R) =(1, 0) <br />
=(0, 1) A (1, 0) (1, 1) <br />
A =(1, 0) A =(1, 1) <br />
y<br />
y<br />
(0, 1)<br />
(1, 1)<br />
(1, 0)<br />
x<br />
(1, 0)<br />
x<br />
SL 2 (R) <br />
O(n)<br />
GL n (R) A A −1 =<br />
A O(n)<br />
n × n O(n) <br />
GL n (R)<br />
<br />
( )<br />
3/5 −4/5<br />
,<br />
4/5 3/5<br />
( √ )<br />
1/2 − 3/2<br />
√ ,<br />
3/2 1/2<br />
⎛<br />
−1/ √ 2 0 1/ √ ⎞<br />
2<br />
⎝ 1/ √ 6 −2/ √ 6 1/ √ 6<br />
1/ √ 3 1/ √ 3 1/ √ ⎠ .<br />
3<br />
O(n) <br />
<br />
<br />
=(x 1 ,...,x n ) =(y 1 ,...,y n ) <br />
⎛ ⎞<br />
y 1<br />
〈, 〉 = y 2<br />
=(x 1 ,x 2 ,...,x n ) ⎜ ⎟<br />
⎝ ⎠ = x 1y 1 + ···+ x n y n .<br />
y n<br />
=(x 1 ,...,x n ) <br />
‖‖ = √ √<br />
〈, 〉 = x 2 1 + ···+ x2 n.<br />
<br />
‖ − ‖
R n α ∈ R <br />
〈, 〉 = 〈, 〉<br />
〈, + 〉 = 〈, 〉 + 〈, 〉<br />
〈α, 〉 = 〈,α〉 = α〈, 〉<br />
〈, 〉 ≥0 =0<br />
〈, 〉 =0 R n =0<br />
=(3, 4) √ 3 2 +4 2 =5 <br />
<br />
( )<br />
3/5 −4/5<br />
A =<br />
4/5 3/5<br />
A =(−7/5, 24/5) <br />
(AA )=(I) =1 (A) =(A ) <br />
1 −1 <br />
⎛ ⎞<br />
a 1j<br />
a 2j<br />
j = ⎜ ⎟<br />
⎝ ⎠<br />
a nj<br />
A =(a ij ) AA = I 〈 r , s 〉 = δ rs <br />
δ rs =<br />
{ 1 r = s<br />
0 r ≠ s<br />
<br />
<br />
n×n <br />
A A −1 = A <br />
A <br />
‖T − T ‖ = ‖ − ‖ ‖T ‖ = ‖‖ 〈T ,T〉 = 〈, 〉 <br />
<br />
<br />
A n × n <br />
A <br />
A −1 = A <br />
〈A,A〉 = 〈, 〉<br />
‖A − A‖ = ‖ − ‖<br />
‖A‖ = ‖‖
(2) ⇒ (3)<br />
(3) ⇒ (2) <br />
〈A,A〉 =(A) A<br />
= A A<br />
= <br />
= 〈, 〉.<br />
〈, 〉 = 〈A,A〉<br />
= A A<br />
= 〈,A A〉,<br />
〈, (A A − I)〉 =0 A A − I =0 A −1 = A <br />
(3) ⇒ (4) A A <br />
‖A − A‖ 2 = ‖A( − )‖ 2<br />
= 〈A( − ),A( − )〉<br />
= 〈 − , − 〉<br />
= ‖ − ‖ 2 .<br />
(4) ⇒ (5) A A =0<br />
<br />
‖A‖ = ‖A − A‖ = ‖ − ‖ = ‖‖.<br />
(5) ⇒ (3) <br />
<br />
〈, 〉 = 1 [<br />
‖ + ‖ 2 −‖‖ 2 −‖‖ 2] .<br />
2<br />
<br />
〈A,A〉 = 1 [<br />
‖A + A‖ 2 −‖A‖ 2 −‖A‖ 2]<br />
2<br />
= 1 [<br />
‖A( + )‖ 2 −‖A‖ 2 −‖A‖ 2]<br />
2<br />
= 1 [<br />
‖ + ‖ 2 −‖‖ 2 −‖‖ 2]<br />
2<br />
= 〈, 〉.<br />
y<br />
y<br />
( θ, − θ)<br />
(a, b)<br />
(a, −b)<br />
x<br />
θ<br />
( θ, θ)<br />
x<br />
O(2) R 2
R 2 <br />
T ∈ O(2) 1 =(1, 0) 2 =(0, 1) T ( 1 )=(a, b) <br />
a 2 + b 2 =1 T ( 2 )=(−b, a) T <br />
( ) ( )<br />
a −b θ − θ<br />
A = =<br />
,<br />
b a θ θ<br />
0 ≤ θ
R n T () = + <br />
T O(n) <br />
R 2 R 2 <br />
<br />
R n<br />
<br />
y<br />
y<br />
<br />
x<br />
x<br />
T ()<br />
<br />
f R 2 <br />
f O(2)<br />
f R 2 f <br />
f(0) = 0 ‖f()‖ = ‖‖ <br />
<br />
‖‖ 2 − 2〈f(),f()〉 + ‖‖ 2 = ‖f()‖ 2 − 2〈f(),f()〉 + ‖f()‖ 2<br />
= 〈f() − f(),f() − f()〉<br />
= ‖f() − f()‖ 2<br />
= ‖ − ‖ 2<br />
= 〈 − , − 〉<br />
= ‖‖ 2 − 2〈, 〉 + ‖‖ 2 .<br />
〈f(),f()〉 = 〈, 〉.<br />
1 2 (1, 0) (0, 1) <br />
<br />
=(x 1 ,x 2 )=x 1 1 + x 2 2 ,<br />
f() =〈f(),f( 1 )〉f( 1 )+〈f(),f( 2 )〉f( 2 )=x 1 f( 1 )+x 2 f( 2 ).<br />
f <br />
f T f R 2 <br />
T f() =A A ∈ O(2) f() =A+ <br />
f() =A + 1<br />
g() =B + 2 ,
f(g()) = f(B + 2 )=AB + A 2 + 1 .<br />
R 2 E(2)<br />
R 2 E(2)<br />
R n R n <br />
X ⊂ R 2 X <br />
R n X R 1 Z 2 <br />
R 2 O(2)<br />
R 2 Z n D n <br />
G = {f 1 ,f 2 ,...,f n } <br />
X ⊂ R 2 ∈ X <br />
G X S = { 1 , 2 ,... n } i = f i () <br />
= 1 n<br />
n∑<br />
i .<br />
X <br />
<br />
G R 2 <br />
O(2) <br />
O(2) <br />
( )<br />
θ − θ<br />
R θ =<br />
θ θ<br />
i=1<br />
<br />
( )( )<br />
φ − φ 1 0<br />
T φ =<br />
φ φ 0 −1<br />
( )<br />
φ φ<br />
=<br />
.<br />
φ − φ<br />
(R θ )=1 (T φ )=−1 Tφ 2 = I <br />
G <br />
G −1<br />
G G <br />
G θ 0 <br />
R θ0 R θ0 <br />
G n θ 1 nθ 0 (n +1)θ 0 <br />
(n +1)θ 0 − θ 1 θ 0 <br />
θ 0 <br />
G T φ : G →<br />
{−1, 1} A ↦→ (A) <br />
|G/ φ| =2 G<br />
n |G| =2n G <br />
R θ ,...,R n−1<br />
θ<br />
,TR θ ,...,TR n−1<br />
θ<br />
.<br />
<br />
TR θ T = R −1<br />
θ .<br />
G D n
R 3 <br />
<br />
<br />
R 2<br />
R 3<br />
<br />
R 2 <br />
m + n m<br />
n <br />
(1, 1) (2, 0) <br />
(−1, 1) (−1, −1) <br />
<br />
{ 1 , 2 } { 1 , 2 } <br />
1 = α 1 1 + α 2 2<br />
2 = β 1 1 + β 2 2 ,<br />
α 1 α 2 β 1 β 2 <br />
( )<br />
α1 α<br />
U = 2<br />
.<br />
β 1 β 2<br />
1 2 1 2 U −1 <br />
( ) ( )<br />
U −1 1 1<br />
= .<br />
2 2
U U −1 <br />
U U −1 UU −1 = I<br />
(UU −1 )=(U) (U −1 )=1;<br />
(U) =±1 ±1 <br />
<br />
( ) 3 1<br />
5 2<br />
<br />
(−1, 1)<br />
(1, 1)<br />
(2, 0)<br />
(−1, −1)<br />
R 2<br />
<br />
E(2) R 2<br />
<br />
R 3 <br />
<br />
<br />
<br />
<br />
<br />
<br />
G ⊂ E(2) {(I,t):t ∈ L} L <br />
<br />
<br />
R 2 Z × Z<br />
G G 0 = {A :(A, b) ∈ G b} G 0 <br />
O(2) L G <br />
H G 0 (A, ) G<br />
(A, )(I,)(A, ) −1 =(A, A + )(A −1 , −A −1 )<br />
=(AA −1 , −AA −1 + A + )<br />
=(I,A);<br />
(I,A) G A <br />
L G 0 G <br />
T G G/T ∼ = G 0
Z n D n <br />
n =1, 2, 3, 4, 6<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
R 2<br />
<br />
<br />
<br />
Z 1 <br />
Z 2 <br />
Z 3 <br />
Z 4 <br />
Z 6 <br />
D 1 <br />
D 1 <br />
D 1 <br />
D 2 <br />
D 2 <br />
D 2 <br />
D 2 <br />
D 3 <br />
D 4 <br />
D 6
p4m<br />
p4g<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
n <br />
<br />
R 3 <br />
R 4 <br />
R 5 <br />
R 3 <br />
<br />
<br />
<br />
<br />
<br />
〈, 〉 = 1 [<br />
‖ + ‖ 2 −‖‖ 2 −‖‖ 2] .<br />
2<br />
<br />
<br />
<br />
O(n) <br />
SO(n)<br />
<br />
( √ √ )<br />
1/ 2 −1/ 2<br />
1/ √ 2 1/ √ 2<br />
( √ √ )<br />
1/ 5 2/ 5<br />
−2/ √ 5 1/ √ 5
⎛<br />
4/ √ 5 0 3/ √ ⎞<br />
5<br />
⎝−3/ √ 5 0 4/ √ 5⎠<br />
0 −1 0<br />
<br />
⎛<br />
1/3 2/3<br />
⎞<br />
−2/3<br />
⎝−2/3 2/3 1/3 ⎠<br />
−2/3 1/3 2/3<br />
<br />
<br />
<br />
<br />
<br />
<br />
R n α ∈ R <br />
<br />
〈, 〉 = 〈, 〉<br />
〈, + 〉 = 〈, 〉 + 〈, 〉<br />
〈α, 〉 = 〈,α〉 = α〈, 〉<br />
〈, 〉 ≥0 =0<br />
〈, 〉 =0 R n =0<br />
<br />
E(n) ={(A, ) :A ∈ O(n) ∈ R n }<br />
<br />
{(2, 1), (1, 1)} {(12, 5), (7, 3)} <br />
G E(2) T G<br />
G G/T <br />
A ∈ SL 2 (R) <br />
R 2 <br />
A A<br />
SO(n) O(n)<br />
f R n <br />
E(2) (A, ) ≠0 <br />
<br />
O(n)<br />
=(x 1 ,x 2 ) R 2 x 2 1 + x2 2 =1A ∈ O(2)<br />
A <br />
G H <br />
N G N H
H ∩ N = {}<br />
HN = G<br />
<br />
S 3 A 3 H = {(1), (12)}<br />
Q 8 <br />
E(2) O(2) H H <br />
R 2
13<br />
<br />
<br />
<br />
n <br />
Z n <br />
<br />
<br />
<br />
<br />
G = H n ⊃ H n−1 ⊃···⊃H 1 ⊃ H 0 = {e},<br />
H i H i+1 H i+1 /H i <br />
G <br />
<br />
<br />
<br />
<br />
<br />
Z p p Z mn<br />
∼ = Zm × Z n <br />
(m, n) =1 <br />
<br />
<br />
Z p<br />
α 1<br />
1<br />
×···×Z p<br />
αn<br />
n ,<br />
p k <br />
G<br />
{g i } G i I <br />
G g i <br />
G g i G G G <br />
{g i : i ∈ I} g i G <br />
{g i : i ∈ I} G G <br />
<br />
S 3 (12) (123) Z × Z n <br />
{(1, 0), (0, 1)}<br />
Q<br />
Q
p 1 /q 1 ,...,p n /q n p i /q i p <br />
q 1 ,...,q n 1/p <br />
Q p 1 /q 1 ,...,p n /q n p <br />
<br />
<br />
p i /q i + p j /q j =(p i q j + p j q i )/(q i q j ).<br />
H G {g i ∈ G : i ∈ I}<br />
h ∈ H <br />
h = g α 1<br />
i 1<br />
g ik <br />
···g αn<br />
i n<br />
,<br />
K g α 1<br />
i 1 ···g αn<br />
i n<br />
g ik <br />
K H K <br />
G K = H H <br />
g i <br />
K gi 0 =1 <br />
K g = g k 1<br />
i 1 ···g kn<br />
i n<br />
K <br />
K <br />
g −1 =(g k 1<br />
i 1 ···g kn<br />
i n<br />
) −1 =(g −kn<br />
i n<br />
···g −k 1<br />
i 1<br />
).<br />
g i <br />
g i <br />
a −3 b 5 a 7 <br />
a 4 b 5 <br />
<br />
p <br />
G p G <br />
p Z 2 × Z 2 Z 4 2 Z 27 3 <br />
<br />
p<br />
<br />
G <br />
Z p<br />
α 1<br />
1<br />
× Z p<br />
α 2<br />
2<br />
p i <br />
×···×Z p<br />
αn<br />
n<br />
540 = 2 2 ·3 3 ·5<br />
<br />
<br />
Z 2 × Z 2 × Z 3 × Z 3 × Z 3 × Z 5 <br />
Z 2 × Z 2 × Z 3 × Z 9 × Z 5 <br />
Z 2 × Z 2 × Z 27 × Z 5 <br />
Z 4 × Z 3 × Z 3 × Z 3 × Z 5 <br />
Z 4 × Z 3 × Z 9 × Z 5
Z 4 × Z 27 × Z 5 <br />
<br />
<br />
G n p n<br />
G p<br />
n =1 <br />
G n k <br />
k
G G i G <br />
i =1,...,n p 0 i =1 1 ∈ G ig ∈ G i <br />
p r i g−1 p r i h ∈ G i p s i <br />
t r s<br />
<br />
(gh) pt i = g<br />
p t i h<br />
p t i =1· 1=1,<br />
G = G 1 G 2 ···G k<br />
G i ∩ G j = {1} i ≠ j g 1 ∈ G 1 <br />
G 2 ,G 3 ,...,G k g 1 = g 2 g 3 ···g k g i ∈ G i g i p α i<br />
<br />
g pα i<br />
i<br />
=1 i =2, 3,...,k g pα 2<br />
2 ···pα k<br />
k<br />
1 =1 g 1 p 1<br />
(p 1 ,p α 2<br />
2 ···pα k<br />
k )=1 g 1 =1 G 1<br />
G 2 ,G 3 ,...,G k <br />
G i ∩ G j = {1} i ≠ j<br />
g ∈ G g 1 ···g k <br />
g i ∈ G i g G <br />
|g| = p β 1<br />
1 pβ 2<br />
2 ···pβ k<br />
k<br />
β 1 ,...,β k a i = |g|/p β i<br />
i a i <br />
b 1 ,...,b k a 1 b 1 + ···+ a k b k =1 <br />
<br />
g = g a 1b 1 +···+a k b k<br />
= g a 1b1<br />
···g a kb k<br />
.<br />
g (a ib i )p β i<br />
i = g b i|g| = e,<br />
g a ib i<br />
G i g i = g a ib i<br />
g = g 1 ···g k ∈ G 1 G 2 ···G k <br />
G = G 1 G 2 ···G k <br />
p i G i <br />
G p g ∈ G <br />
G 〈g〉×H H G<br />
G p n <br />
n n =1 G p g <br />
k 1 ≤ k
a p a /∈ 〈g〉 h <br />
〈g〉 |H| = p<br />
gH G/H <br />
g G |gH| < |g| = p m <br />
H =(gH) pm−1 = g pm−1 H;<br />
g pm−1 〈g〉 ∩H = {e} g <br />
p m gH G/H <br />
<br />
G/H ∼ = 〈gH〉×K/H<br />
K G H 〈g〉∩K = {e} b ∈〈g〉∩K <br />
bH ∈〈gH〉∩K/H = {H} b ∈〈g〉∩H = {e} G = 〈g〉K <br />
G ∼ = 〈g〉×K<br />
<br />
G g <br />
G 〈g〉 = G G ∼ = Z |g| × H <br />
H G |H| < |G| <br />
<br />
<br />
<br />
<br />
G <br />
<br />
Z p<br />
α 1<br />
1<br />
× Z p<br />
α 2<br />
2<br />
×···×Z p<br />
αn<br />
n<br />
p i <br />
× Z ×···×Z,<br />
<br />
<br />
G <br />
G = H n ⊃ H n−1 ⊃···⊃H 1 ⊃ H 0 = {e},<br />
H i H i+1 H i G <br />
<br />
<br />
<br />
<br />
Z ⊃ 9Z ⊃ 45Z ⊃ 180Z ⊃{0},<br />
Z 24 ⊃〈2〉 ⊃〈6〉 ⊃〈12〉 ⊃{0}.<br />
<br />
D 4 <br />
D 4 ⊃{(1), (12)(34), (13)(24), (14)(23)} ⊃{(1), (12)(34)} ⊃{(1)}.<br />
{(1), (12)(34)} D 4
{K j } <br />
{H i } {H i }⊂{K j } H i K j <br />
<br />
<br />
Z ⊃ 3Z ⊃ 9Z ⊃ 45Z ⊃ 90Z ⊃ 180Z ⊃{0}<br />
Z ⊃ 9Z ⊃ 45Z ⊃ 180Z ⊃{0}.<br />
{H i } G <br />
H i+1 /H i {H i } <br />
{K j } G <br />
{H i+1 /H i } {K j+1 /K j }<br />
<br />
Z 60 <br />
Z 60 ⊃〈3〉 ⊃〈15〉 ⊃{0}<br />
Z 60 ⊃〈4〉 ⊃〈20〉 ⊃{0}<br />
Z 60 /〈3〉 ∼ = 〈20〉/{0} ∼ = Z 3<br />
〈3〉/〈15〉 ∼ = 〈4〉/〈20〉 ∼ = Z 5<br />
〈15〉/{0} ∼ = Z 60 /〈4〉 ∼ = Z 4 .<br />
{H i } G <br />
<br />
{H i } G <br />
Z 60 <br />
<br />
Z 60 ⊃〈3〉 ⊃〈15〉 ⊃〈30〉 ⊃{0}<br />
Z 60 /〈3〉 ∼ = Z 3<br />
〈3〉/〈15〉 ∼ = Z 5<br />
〈15〉/〈30〉 ∼ = Z 2<br />
〈30〉/{0} ∼ = Z 2 .<br />
Z 60 <br />
<br />
<br />
n ≥ 5 <br />
Z 60 ⊃〈2〉 ⊃〈4〉 ⊃〈20〉 ⊃{0}<br />
S n ⊃ A n ⊃{(1)}<br />
S n S n /A n<br />
∼ = Z2 A n
{0} = H 0 ⊂ H 1 ⊂···⊂H n−1 ⊂ H n = Z<br />
H 1 kZ<br />
k ∈ N H 1 /H 0<br />
∼ = kZ <br />
<br />
Z 60 <br />
<br />
Z 60 Z 2 Z 2 Z 3 Z 5 <br />
<br />
G <br />
<br />
G <br />
<br />
<br />
k 1 ≤ k
H n−1 H n−1 K m−1 = G <br />
<br />
<br />
<br />
K m−1 /(K m−1 ∩ H n−1 ) ∼ = (H n−1 K m−1 )/H n−1 = G/H n−1 .<br />
G = H n ⊃ H n−1 ⊃ H n−1 ∩ K m−1 ⊃···⊃H 0 ∩ K m−1 = {e}<br />
G = K m ⊃ K m−1 ⊃ K m−1 ∩ H n−1 ⊃···⊃K 0 ∩ H n−1 = {e}<br />
<br />
G {H i } <br />
H i+1 /H i <br />
<br />
S 4 <br />
S 4 ⊃ A 4 ⊃{(1), (12)(34), (13)(24), (14)(23)} ⊃{(1)}<br />
n ≥ 5 <br />
S n ⊃ A n ⊃{(1)}<br />
S n S n <br />
n ≥ 5<br />
<br />
<br />
<br />
16<br />
<br />
<br />
<br />
<br />
<br />
<br />
40 <br />
200 <br />
720 <br />
<br />
Z 12<br />
S 3 × Z 4<br />
Z 48<br />
S 4<br />
Q 8<br />
S n n ≥ 5<br />
D 4 Q<br />
G = Z 2 × Z 2 ×··· <br />
G m n m G <br />
n<br />
G G
G H K G × H ∼ = G × K<br />
H ∼ = K <br />
G H G × H <br />
G N G<br />
N<br />
N G N G/N <br />
G <br />
N G N G/N G<br />
<br />
G G <br />
G = P n ⊃ P n−1 ⊃···⊃P 1 ⊃ P 0 = {e}<br />
P i P i+1 P i+1 /P i <br />
G G <br />
G N G G/N <br />
<br />
D n n<br />
G N G <br />
N G/N <br />
G p H K H <br />
K K H<br />
G n ≥ 2 G <br />
<br />
G ′ G <br />
G a −1 b −1 ab a, b ∈ G <br />
G G (0) = G G (1) = G ′ G (i+1) =(G (i) ) ′ <br />
G (i+1) (G (i) ) ′ <br />
G<br />
G (0) = G ⊃ G (1) ⊃ G (2) ⊃···<br />
G G (n) = {e} n<br />
G n ≥ 2 G <br />
<br />
H K G <br />
H ∗ K ∗ H K <br />
H ∗ (H ∩ K ∗ ) H ∗ (H ∩ K)<br />
K ∗ (H ∗ ∩ K) K ∗ (H ∩ K)<br />
H ∗ (H ∩ K)/H ∗ (H ∩ K ∗ ) ∼ = K ∗ (H ∩ K)/K ∗ (H ∗ ∩ K) ∼ = (H ∩ K)/(H ∗ ∩ K)(H ∩ K ∗ )<br />
<br />
G
n <br />
n
14<br />
<br />
G X <br />
g ∈ G x ∈ X gx X <br />
<br />
<br />
<br />
<br />
<br />
X G G X G × X → X <br />
(g, x) ↦→ gx <br />
ex = x x ∈ X<br />
(g 1 g 2 )x = g 1 (g 2 x) x ∈ X g 1 ,g 2 ∈ G<br />
X G X <br />
G G X <br />
(g, x) ↦→ x X <br />
G<br />
G = GL 2 (R) X = R 2 G X <br />
v ∈ R 2 I Iv = v A B 2×2 <br />
(AB)v = A(Bv) <br />
G = D 4 X = {1, 2, 3, 4}<br />
D 4 <br />
<br />
{(1), (13), (24), (1432), (1234), (12)(34), (14)(23), (13)(24)}.<br />
D 4 X (13)(24) 1 <br />
3 2 4 <br />
<br />
X G S X <br />
X X G <br />
σ ∈ G x ∈ X<br />
(σ, x) ↦→ σ(x)
X = G G <br />
(g, x) ↦→ λ g (x) =gx λ g <br />
e · x = λ e x = ex = x<br />
(gh) · x = λ gh x = λ g λ h x = λ g (hx) =g · (h · x).<br />
H G G H H<br />
G X = G H G <br />
G H H G<br />
<br />
H × G → G,<br />
(h, g) ↦→ hgh −1<br />
h ∈ H g ∈ G <br />
(h 1 h 2 ,g)=h 1 h 2 g(h 1 h 2 ) −1<br />
<br />
= h 1 (h 2 gh −1<br />
2 )h−1 1<br />
=(h 1 , (h 2 ,g)),<br />
H G L H H L H<br />
G <br />
(g, xH) ↦→ gxH.<br />
(gg ′ )xH = g(g ′ xH) <br />
<br />
G X x, y ∈ X x G y <br />
g ∈ G gx = y x ∼ G y x ∼ y G<br />
X G G X<br />
∼ ex = x x ∼ y x, y ∈ X <br />
g gx = y g −1 y = x y ∼ x <br />
x ∼ y y ∼ z <br />
g h gx = y hy = z z = hy =(hg)x x z<br />
X G X G <br />
X G x X O x <br />
G <br />
G = {(1), (123), (132), (45), (123)(45), (132)(45)}<br />
X = {1, 2, 3, 4, 5} X G O 1 = O 2 = O 3 = {1, 2, 3} <br />
O 4 = O 5 = {4, 5}<br />
G X g G <br />
g X X g x ∈ X gx = x <br />
g x ∈ X G<br />
<br />
x x G x
X g ⊂ X G x ⊂ G<br />
X = {1, 2, 3, 4, 5, 6} G <br />
<br />
{(1), (12)(3456), (35)(46), (12)(3654)}.<br />
X G <br />
<br />
X (1) = X,<br />
X (35)(46) = {1, 2},<br />
X (12)(3456) = X (12)(3654) = ∅,<br />
G 1 = G 2 = {(1), (35)(46)},<br />
G 3 = G 4 = G 5 = G 6 = {(1)}.<br />
G x G x ∈ X<br />
G X x ∈ X <br />
x G x G<br />
e ∈ G x X g, h ∈ G x <br />
gx = x hx = x (gh)x = g(hx) =gx = x <br />
G x G x g ∈ G x x = ex =(g −1 g)x =(g −1 )gx = g −1 x g −1<br />
G x <br />
g ∈ G <br />
|X g | x ∈ X |O x | <br />
x ∈ X G x<br />
G<br />
G X G x ∈ X |O x | =[G :<br />
G x ]<br />
|G|/|G x | G x G <br />
φ O x X <br />
L Gx G x G y ∈O x g G <br />
gx = y φ φ(y) =gG x φ φ(y 1 )=φ(y 2 )<br />
<br />
φ(y 1 )=g 1 G x = g 2 G x = φ(y 2 ),<br />
g 1 x = y 1 g 2 x = y 2 g 1 G x = g 2 G x g ∈ G x g 2 = g 1 g<br />
y 2 = g 2 x = g 1 gx = g 1 x = y 1 ;<br />
φ φ <br />
gG x gx = y φ(y) =gG x
X G X G X <br />
X<br />
X G = {x ∈ X : gx = x g ∈ G}.<br />
|X| = |X G | +<br />
n∑<br />
|O xi |,<br />
x k ,...,x n X<br />
G (g, x) ↦→ gxg −1 <br />
G<br />
Z(G) ={x : xg = gx g ∈ G},<br />
<br />
G x 1 ,...,x k <br />
G |O x1 | = n 1 ,...,|O xk | = n k <br />
i=k<br />
|G| = |Z(G)| + n 1 + ···+ n k .<br />
x i C(x i )={g ∈ G : gx i = x i g} <br />
x i <br />
|G| = |Z(G)| +[G : C(x 1 )] + ···+[G : C(x k )].<br />
<br />
G<br />
S 3 <br />
6=1+2+3<br />
{(1)}, {(123), (132)}, {(12), (13), (23)}.<br />
D 4 {(1), (13)(24)} <br />
{(13), (24)}, {(1432), (1234)}, {(12)(34), (14)(23)}.<br />
D 4 8=2+2+2+2<br />
S n <br />
σ =(a 1 ,...,a k ) τ ∈ S n <br />
τστ −1 =(τ(a 1 ),...,τ(a k )).<br />
σ = σ 1 σ 2 ···σ r <br />
σ i r i σ <br />
τ ∈ S n <br />
S n n <br />
S 3 <br />
3 <br />
3=1+1+1<br />
3=1+2
3=3;<br />
<br />
n <br />
n <br />
<br />
G p n p G <br />
<br />
<br />
<br />
|G| = |Z(G)| + n 1 + ···+ n k .<br />
n i > 1 n i ||G| p n i p ||G| p<br />
|Z(G)| G |Z(G)| ≥1 <br />
|Z(G)| ≥p g ∈ Z(G) g ≠1<br />
G p 2 p G <br />
|Z(G)| = p p 2 |Z(G)| = p 2 <br />
|Z(G)| = p Z(G) G/Z(G) p <br />
aZ(G) G/Z(G) gZ(G) <br />
a m Z(G) m g = a m x x <br />
G hZ(G) ∈ G/Z(G) y Z(G) h = a n y <br />
n x y G G<br />
<br />
gh = a m xa n y = a m+n xy = a n ya m x = hg,<br />
G <br />
<br />
<br />
<br />
2 4 =16 <br />
<br />
<br />
90 ◦ <br />
B<br />
W<br />
W<br />
B<br />
W<br />
W<br />
W<br />
W<br />
W<br />
W<br />
W<br />
W<br />
B<br />
W<br />
W<br />
B
X G x ∼ y G x G y <br />
|G x | = |G y |<br />
G X (g, x) ↦→ g · x x ∼ y g ∈ G <br />
g · x = y a ∈ G x <br />
gag −1 · y = ga · g −1 y = ga · x = g · x = y,<br />
φ : G x → G y φ(a) =gag −1 φ <br />
φ(ab) =gabg −1 = gag −1 gbg −1 = φ(a)φ(b).<br />
φ(a) =φ(b) gag −1 = gbg −1 a = b <br />
φ b G y g −1 bg G x <br />
φ(g −1 bg) =b<br />
g −1 bg · x = g −1 b · gx = g −1 b · y = g −1 · y = x;<br />
G X k <br />
X <br />
k = 1 ∑<br />
|X g |.<br />
|G|<br />
g∈G<br />
x g ∈ G <br />
g x gx = x <br />
g x <br />
∑<br />
|X g |.<br />
<br />
∑<br />
|G x |;<br />
g∈G<br />
x∈X<br />
∑ g∈G |X g| = ∑ x∈X |G x| <br />
∑<br />
|G y | = |O x |·|G x |.<br />
y∈O x<br />
|G| <br />
k <br />
∑<br />
|X g | = ∑ |G x | = k ·|G|.<br />
g∈G x∈X<br />
X = {1, 2, 3, 4, 5} G <br />
G = {(1), (13), (13)(25), (25)} X {1, 3} {2, 5} {4} <br />
<br />
X (1) = X
k = 1<br />
|G|<br />
X (13) = {2, 4, 5}<br />
X (13)(25) = {4}<br />
X (25) = {1, 3, 4}.<br />
∑<br />
|X g | = 1 4 (5+3+1+3)=3.<br />
g∈G<br />
<br />
<br />
<br />
<br />
<br />
<br />
D 4 <br />
(1) (13) (24) (1432)<br />
(1234) (12)(34) (14)(23) (13)(24)<br />
G {1, 2, 3, 4} <br />
X Y = {B,W} B W <br />
f : X → Y <br />
σ ∈ D 4 ˜σ <br />
˜σ(f) =f ◦ σ f : X → Y f <br />
f(1) = B<br />
f(2) = W<br />
f(3) = W<br />
f(4) = W<br />
σ = (12)(34) ˜σ(f) =f ◦ σ 2 B <br />
W ˜σ ˜G ˜X<br />
˜X X <br />
Y ˜G <br />
<br />
˜X(1) = ˜X | ˜X| =2 4 = 16<br />
<br />
˜X(1234) f ∈ ˜X f (1234) <br />
f(1) = f(2) = f(3) = f(4) f <br />
f(x) =B f(x) =W x | ˜X (1234) | =2<br />
| ˜X (1432) | =2<br />
˜X (13)(24) f(1) = f(3) f(2) = f(4) | ˜X (13)(24) | =2 2 =4<br />
| ˜X (12)(34) | =4<br />
| ˜X (14)(23) | =4
˜X (13) f(1) = f(3) | ˜X (13) | =<br />
2 3 =8<br />
| ˜X (24) | =8<br />
<br />
<br />
1<br />
8 (24 +2 1 +2 2 +2 1 +2 2 +2 2 +2 3 +2 3 )=6<br />
G X ˜X <br />
X Y ˜G ˜X ˜σ ∈ ˜G <br />
˜σ(f) =f ◦ σ σ ∈ G f ∈ ˜X n <br />
σ | ˜X σ | = |Y | n <br />
σ ∈ G f ∈ ˜X f ◦ σ ˜X g <br />
X Y ˜σ(f) =˜σ(g) x ∈ X<br />
f(σ(x)) = ˜σ(f)(x) =˜σ(g)(x) =g(σ(x)).<br />
σ X x ′ X x X σ<br />
f g X f = g ˜σ <br />
σ ↦→ ˜σ <br />
σ X σ = σ 1 σ 2 ···σ n <br />
f ˜X σ σ n |Y |<br />
| ˜X σ | = |Y | n <br />
X = {1, 2,...,7} Y = {A, B, C} g <br />
X (13)(245) = (13)(245)(6)(7) n =4 f ∈ ˜X g <br />
g |Y | =3 <br />
| ˜X g | =3 4 =81<br />
<br />
<br />
<br />
1<br />
8 (44 +4 1 +4 2 +4 1 +4 2 +4 2 +4 3 +4 3 )=55<br />
<br />
<br />
n
x 1<br />
x 2<br />
<br />
f f(x 1 ,x 2 ,...,x n )<br />
x n<br />
n <br />
n Z n 2<br />
Z 2 n<br />
2 n n 2 2n n <br />
<br />
<br />
a<br />
b<br />
f f(a, b)<br />
a<br />
b<br />
f f(b, a) =g(a, b)<br />
<br />
a b <br />
f g g f<br />
g(a, b, c) =f(b, c, a) <br />
g ∼ f (acb) <br />
(ab) <br />
f 2 ∼ f 4<br />
f 3 ∼ f 5<br />
f 10 ∼ f 12<br />
f 11 ∼ f 13 .<br />
<br />
<br />
f 0 f 1 f 2 f 3 f 4 f 5 f 6 f 7<br />
<br />
<br />
<br />
<br />
<br />
<br />
f 8 f 9 f 10 f 11 f 12 f 13 f 14 f 15<br />
<br />
<br />
<br />
<br />
<br />
2 23 = 256 <br />
2 24 =65,536
a b c <br />
f g <br />
f g <br />
{a, b, c} <br />
a b c <br />
2 3 <br />
(a, b, c) (acb) <br />
<br />
(0, 0, 0) ↦→ (0, 0, 0)<br />
(0, 0, 1) ↦→ (0, 1, 0)<br />
(0, 1, 0) ↦→ (1, 0, 0)<br />
<br />
(1, 1, 0) ↦→ (1, 0, 1)<br />
(1, 1, 1) ↦→ (1, 1, 1).<br />
X n |X| =2 n <br />
<br />
(0,...,0, 1) ↦→ 0<br />
(0,...,1, 0) ↦→ 1<br />
(0,...,1, 1) ↦→ 2<br />
<br />
(1,...,1, 1) ↦→ 2 n − 1.<br />
<br />
<br />
(a), (ac), (bd), (adcb),<br />
(abcd), (ab)(cd), (ad)(bc), (ac)(bd).<br />
<br />
<br />
<br />
1<br />
8 (216 +2· 2 12 +2· 2 6 +3· 2 10 ) = 9616
(a) (0) <br />
(ac) (2, 8)(3, 9)(6, 12)(7, 13) <br />
(bd) (1, 4)(3, 6)(9, 12)(11, 14) <br />
(adcb) (1, 2, 4, 8)(3, 6.12, 9)(5, 10)(7, 14, 13, 11) <br />
(abcd) (1, 8, 4, 2)(3, 9, 12, 6)(5, 10)(7, 11, 13, 14) <br />
(ab)(cd) (1, 2)(4, 8)(5, 10)(6, 9)(7, 11)(13, 14) <br />
(ad)(bc) (1, 8)(2, 4)(3, 12)(5, 10)(7, 14)(11, 13) <br />
(ac)(bd) (1, 4)(2, 8)(3, 12)(6, 9)(7, 13)(11, 14) <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
G <br />
<br />
G = H n ⊃ H n−1 ⊃···⊃H 1 ⊃ H 0 = {e}<br />
H i H i+1 H i+1 /H i <br />
<br />
<br />
<br />
<br />
<br />
G <br />
X G <br />
G<br />
<br />
X g G x <br />
X = {1, 2, 3} G = S 3 = {(1), (12), (13), (23), (123), (132)}<br />
X = {1, 2, 3, 4, 5, 6} G = {(1), (12), (345), (354), (12)(345), (12)(354)}<br />
G X G <br />
x ∈ X |G| = |O x |·|G x |
G θ ∈ G <br />
R 2 θ <br />
P <br />
R 2 G<br />
P <br />
G P <br />
<br />
G = A 4 G (g, h) ↦→ ghg −1 <br />
G<br />
G<br />
<br />
<br />
S 4 D 5 Z 9 Q 8<br />
S 5 A 5 <br />
<br />
<br />
<br />
<br />
<br />
1,...,6 <br />
<br />
<br />
12 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
CH 3
H<br />
H<br />
H<br />
H<br />
H<br />
H<br />
<br />
<br />
x 1 x 2 x 3 S 3 x 1 <br />
x 2 x 3 x 4 S 4 <br />
<br />
x 1 x 2 x 3 x 4 S 4 <br />
(x 1 x 2 x 3 x 4 )<br />
12 <br />
<br />
G X G <br />
X G X <br />
G X<br />
p p n <br />
S n <br />
a ∈ G g ∈ G gC(a)g −1 = C(gag −1 )<br />
|G| = p n p |Z(G)|
15<br />
<br />
G <br />
m n m G n <br />
A 4 12 6 <br />
<br />
<br />
<br />
<br />
<br />
<br />
G <br />
<br />
G (g, x) ↦→ gxg −1 x 1 ,...,x k <br />
G <br />
<br />
|G| = |Z(G)| +[G : C(x 1 )] + ···+[G : C(x k )],<br />
Z(G) ={g ∈ G : gx = xg x ∈ G} G C(x i )={g ∈ G :<br />
gx i = x i g} x i <br />
p<br />
p G p G <br />
p p G p p<br />
G p p <br />
G G p<br />
G |G| = p G <br />
k p ≤ k
G Z(G) Z(G) p <br />
G <br />
p<br />
G G p |G| = p n <br />
A 5 |A 5 | =60=2 2 · 3 · 5 <br />
A 5 2 3 5 <br />
A 5 <br />
<br />
<br />
G p <br />
p r |G| G p r <br />
G |G| = p <br />
G n n>p <br />
n p n <br />
|G| = |Z(G)| +[G : C(x 1 )] + ···+[G : C(x k )].<br />
p [G : C(x i )] i p r ||C(x i )| p r<br />
|G| = |C(x i )|·[G : C(x i )] C(x i )<br />
p [G : C(x i )] i p |G| <br />
p |Z(G)| Z(G) <br />
p g N g N <br />
Z(G) Z(G) N G Z(G)<br />
G G/N |G|/p <br />
G/N H p r−1 <br />
H φ : G → G/N p r G<br />
p P G p G <br />
<br />
G S G <br />
H S H H S <br />
H ×S →S<br />
<br />
h · K ↦→ hKh −1<br />
K S<br />
<br />
N(H) ={g ∈ G : gHg −1 = H}<br />
G H G H <br />
N(H) N(H) G H <br />
P p G x <br />
p x −1 Px = P x ∈ P <br />
x ∈ N(P ) 〈xP 〉⊂N(P )/P <br />
p H N(P ) <br />
P H/P = 〈xP 〉 |H| = |P |·|〈xP 〉| H <br />
p P p H P <br />
p |G| H = P H/P xP = P <br />
x ∈ P
H K G H <br />
K [H : N(K) ∩ H]<br />
K <br />
N(K) ∩ H h −1 Kh ↦→ (N(K) ∩ H)h h 1 ,h 2 ∈ H<br />
(N(K) ∩ H)h 1 =(N(K) ∩ H)h 2 h 2 h −1<br />
1 ∈ N(K) <br />
K = h 2 h −1<br />
1 Kh 1h −1<br />
2 h −1<br />
1 Kh 1 = h −1<br />
2 Kh 2 <br />
<br />
H K N(K) ∩ H H<br />
G p <br />
|G| p G P 1 P 2 <br />
p g ∈ G gP 1 g −1 = P 2 <br />
<br />
<br />
P p G |G| = p r m |P | = p r <br />
S = {P = P 1 ,P 2 ,...,P k }<br />
P G k =[G : N(P )] <br />
|G| = p r m = |N(P )|·[G : N(P )] = |N(P )|·k.<br />
p r |N(P )| p k<br />
p Q Q ∈S Q<br />
P i S <br />
P i [Q : N(P i ) ∩ Q] <br />
|Q| =[Q : N(P i ) ∩ Q]|N(P i ) ∩ Q| [Q : N(P i ) ∩ Q] |Q| = p r <br />
<br />
p p k <br />
p P j x −1 P j x = P j x ∈ Q <br />
P j = Q<br />
G p <br />
G p 1( p)<br />
|G|<br />
<br />
P p p<br />
S = {P = P 1 ,P 2 ,...,P k },<br />
P P<br />
P p P <br />
p |S| {P } <br />
|S| p 1 |S| ≡ 1( p)<br />
G S p <br />
P ∈S<br />
|S| = | P | =[G : N(P )]<br />
[G : N(P )] |G| <br />
p
A 5 <br />
2 3 4 5 p A 5 3 4 5 <br />
p A 5 <br />
5 60 1( 5) <br />
5 A 5 5 <br />
5 <br />
A 5 A 5 <br />
5 A 5 <br />
<br />
<br />
<br />
p q p
5 · 7 · 47 = 1645 <br />
G ′ = 〈aba −1 b −1 : a, b ∈ G〉 <br />
aba −1 b −1 G G ′ <br />
G G/G ′ <br />
G ′ G G <br />
<br />
5 · 7 · 47 = 1645 <br />
G H 1<br />
47 G/H 1 <br />
G H |G ′ | 1 47 <br />
|G ′ | =1 |G ′ | =47 <br />
G 5 7 <br />
H 2 H 3 G |H 2 | =5 |H 3 | =7 <br />
G ′ H i i =1, 2 G ′ 1<br />
5 7 |G ′ | =1 47 <br />
G G <br />
<br />
<br />
<br />
A 5 <br />
<br />
<br />
G 20 <br />
G 5 <br />
1( 5) 20 1<br />
5 5 <br />
<br />
G p n n>1 p <br />
G G G <br />
4 8 9 16 25 27 32 49 64 81 <br />
4 9 25 49 <br />
56 = 2 3 · 7 <br />
p p <br />
<br />
7 7 <br />
7 <br />
<br />
8 · 6=48 <br />
7 2 <br />
2 2 <br />
2 48 7 <br />
7 2 8 <br />
2 56 2
G G G <br />
48 <br />
48 <br />
H K G <br />
|HK| = |H|·|K|<br />
|H ∩ K| .<br />
<br />
HK = {hk : h ∈ H, k ∈ K}.<br />
|HK|≤|H|·|K| HK <br />
H K h 1 k 1 = h 2 k 2 h 1 ,h 2 ∈ H <br />
k 1 ,k 2 ∈ K <br />
a =(h 1 ) −1 h 2 = k 1 (k 2 ) −1 .<br />
a ∈ H ∩ K (h 1 ) −1 h 2 H k 2 (k 1 ) −1 K <br />
h 2 = h 1 a −1<br />
k 2 = ak 1 .<br />
h = h 1 b −1 k = bk 1 b ∈ H ∩ K hk = h 1 k 1 h ∈ H<br />
k ∈ K hk ∈ HK h i k i h i ∈ H <br />
k i ∈ K H ∩ K |H ∩ K| <br />
|HK| =(|H|·|K|)/|H ∩ K|<br />
G 48 <br />
G 8 16 <br />
G 2 16 <br />
<br />
2 H <br />
K |H ∩ K| =8|H ∩ K| ≤4 <br />
16 · 16<br />
|HK| = =64,<br />
4<br />
H ∩ K H K <br />
H K H ∩ K H <br />
N(H ∩ K) N(H ∩ K) 16 |N(H ∩ K)| <br />
16 1 48 <br />
|N(H ∩ K)| =48 N(H ∩ K) =G<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
p p <br />
p
p G 18 24 54 72 <br />
80<br />
3 S 4 <br />
45 9<br />
H p G H p G<br />
N(H)<br />
96 <br />
160 <br />
H G |H| = p k p <br />
H p G<br />
G p 2 q 2 p q q ∤ p 2 − 1<br />
p ∤ q 2 − 1 G <br />
33 3<br />
H G H <br />
G<br />
G p <br />
p G G<br />
G p r p G <br />
p r−1 <br />
G p n k kq G<br />
<br />
H G <br />
[G : N(H)]<br />
2 S 5 D 4 <br />
<br />
p p m <br />
( p k )<br />
m<br />
p ∤<br />
p k .
S p k G p |S|<br />
G S aT = {at : t ∈ T } a ∈ G <br />
T ∈S <br />
p ∤ |O T | T ∈S<br />
{T 1 ,...,T u } p ∤ u H = {g ∈ G : gT 1 = T 1 } <br />
H G |G| = u|H|<br />
p k |H| p k ≤|H|<br />
|H| = |O T |≤p k p k = |H|<br />
G G ′ = 〈aba −1 b −1 : a, b ∈ G〉 G<br />
G/G ′ {aba −1 b −1 : a, b ∈ G} <br />
<br />
<br />
<br />
<br />
<br />
60 <br />
<br />
<br />
<br />
<br />
1 16 14 31 1 46 2<br />
2 17 1 32 51 47 1<br />
3 18 33 1 48 52<br />
4 19 34 49 <br />
5 20 5 35 1 50 5<br />
6 21 36 14 51 <br />
7 22 2 37 1 52 <br />
8 23 1 38 53 <br />
9 24 39 2 54 15<br />
10 25 2 40 14 55 2<br />
11 26 2 41 1 56 <br />
12 5 27 5 42 57 2<br />
13 28 43 1 58 <br />
14 29 1 44 4 59 1<br />
15 1 30 4 45 60 13<br />
G |G| ≤60<br />
G |G| ≤60 <br />
<br />
G G n n =1,...,60<br />
n
16<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
R <br />
<br />
a + b = b + a a, b ∈ R<br />
(a + b)+c = a +(b + c) a, b, c ∈ R<br />
0 R a +0=a a ∈ R<br />
a ∈ R −a R a +(−a) =0<br />
(ab)c = a(bc) a, b, c ∈ R<br />
a, b, c ∈ R<br />
a(b + c) =ab + ac<br />
(a + b)c = ac + bc.<br />
<br />
<br />
(R, +)<br />
<br />
1 ∈ R 1 ≠0 1a = a1 =a a ∈ R<br />
R R ab = ba a, b R<br />
R <br />
a, b ∈ R ab =0 a =0 b =0 <br />
R R <br />
a ∈ R a ≠0 a −1 a −1 a = aa −1 =1
Z <br />
ab =0 a b a =0 b =0<br />
Z 2 <br />
1/2 1 −1<br />
<br />
Q R <br />
C <br />
a b Z n ab ( n)<br />
Z 12 5 · 7 ≡ 11 ( 12) Z n <br />
Z n <br />
3 · 4 ≡ 0( 12) Z 12 <br />
<br />
a R <br />
b R ab =0 3 4 Z 12 <br />
[a, b] <br />
<br />
f(x) =x 2 g(x) = x (f + g)(x) =f(x)+g(x) =x 2 + x<br />
(fg)(x) =f(x)g(x) =x 2 x<br />
2×2 R <br />
<br />
AB ≠ BA AB =0 A B <br />
<br />
( ) ( ) ( ) ( )<br />
1 0<br />
0 1 0 i<br />
i 0<br />
1= , = , = , = ,<br />
0 1 −1 0 i 0<br />
0 −i<br />
i 2 = −1 <br />
2 = 2 = 2 = −1<br />
= <br />
= <br />
= <br />
= −
= −<br />
= −.<br />
H a + b + c + d a, b, c, d <br />
H 2 × 2 <br />
( ) α β<br />
,<br />
−β α<br />
α = a + di β = b + ci <br />
H 1<br />
<br />
(a 1 + b 1 + c 1 + d 1 )+(a 2 + b 2 + c 2 + d 2 )<br />
=(a 1 + a 2 )+(b 1 + b 2 ) +(c 1 + c 2 ) +(d 1 + d 2 )<br />
<br />
<br />
(a 1 + b 1 + c 1 + d 1 )(a 2 + b 2 + c 2 + d 2 )=α + β + γ + δ,<br />
α = a 1 a 2 − b 1 b 2 − c 1 c 2 − d 1 d 2<br />
β = a 1 b 2 + a 2 b 1 + c 1 d 2 − d 1 c 2<br />
γ = a 1 c 2 − b 1 d 2 + c 1 a 2 + d 1 b 2<br />
δ = a 1 d 2 + b 1 c 2 − c 1 b 2 + d 1 a 2 .<br />
<br />
H <br />
H <br />
<br />
<br />
<br />
(a + b + c + d)(a − b − c − d) =a 2 + b 2 + c 2 + d 2 .<br />
a b c d a + b + c + d ≠0<br />
( )<br />
a − b − c − d<br />
(a + b + c + d)<br />
a 2 + b 2 + c 2 + d 2 =1.<br />
R a, b ∈ R <br />
a0 =0a =0<br />
a(−b) =(−a)b = −ab<br />
(−a)(−b) =ab<br />
<br />
<br />
a0 =a(0 + 0) = a0+a0;<br />
a0 =0 0a =0 ab + a(−b) =a(b − b) =a0 =0<br />
−ab = a(−b) −ab =(−a)b <br />
(−a)(−b) =−(a(−b)) = −(−ab) =ab
S R S R S <br />
R<br />
nZ Z <br />
<br />
<br />
Z ⊂ Q ⊂ R ⊂ C.<br />
<br />
<br />
<br />
R S R S R <br />
<br />
S ≠ ∅<br />
rs ∈ S r, s ∈ S<br />
r − s ∈ S r, s ∈ S<br />
R = M 2 (R) 2 × 2 R T <br />
R <br />
{( )<br />
}<br />
a b<br />
T = : a, b, c ∈ R ,<br />
0 c<br />
T R <br />
( )<br />
( a b<br />
a<br />
′<br />
b<br />
A = B =<br />
′ )<br />
0 c<br />
0 c ′<br />
T A − B T <br />
( aa<br />
′<br />
ab<br />
AB =<br />
′ + bc ′ )<br />
0 cc ′<br />
T <br />
<br />
<br />
R r R <br />
r s ∈ R rs =0<br />
<br />
a R a <br />
R R <br />
<br />
i 2 = −1 Z[i] ={m + ni : m, n ∈ Z} <br />
<br />
α = a + bi<br />
Z[i] α = a − bi αβ =1 αβ =1β = c + di<br />
<br />
1=αβαβ =(a 2 + b 2 )(c 2 + d 2 ).<br />
a 2 + b 2 1 −1 a + bi = ±1 a + bi = ±i<br />
±1 ±i
{( ) 1 0<br />
F = ,<br />
0 1<br />
Z 2 <br />
( ) 1 1<br />
,<br />
1 0<br />
( ) 0 1<br />
,<br />
1 1<br />
( )} 0 0<br />
0 0<br />
Q( √ 2) = {a + b √ 2:a, b ∈ Q} <br />
a + b √ 2 Q( √ 2) <br />
a<br />
a 2 − 2b 2 + −b √<br />
2.<br />
a 2 − 2b 2<br />
<br />
D <br />
D a ∈ D ab = ac <br />
b = c<br />
D D ab = ac <br />
a ≠0 a(b − c) =0 b − c =0 b = c<br />
D <br />
ab = ac b = c ab =0a ≠0 ab = a0 b =0 a <br />
<br />
<br />
<br />
D D ∗ D<br />
D ∗ a ∈ D ∗ <br />
λ a : D ∗ → D ∗ λ a (d) =ad a ≠0 d ≠0 <br />
ad ≠0 λ a d 1 ,d 2 ∈ D ∗ <br />
ad 1 = λ a (d 1 )=λ a (d 2 )=ad 2<br />
d 1 = d 2 D ∗ λ a <br />
d ∈ D ∗ λ a (d) =ad =1 a D <br />
d a D <br />
n r R r + ···+ r n<br />
nr R n<br />
nr =0 r ∈ R R <br />
0 R R<br />
p Z p p <br />
Z p Z p a <br />
pa =0 Z p <br />
p<br />
R 1 n <br />
R n<br />
1 n n n1 =0 <br />
r ∈ R<br />
nr = n(1r) =(n1)r =0r =0.<br />
n n1 =0 R
D D n <br />
n ≠0 n n = ab 1
φ : R → S <br />
R φ(R) <br />
φ(0) = 0<br />
1 R 1 S R S φ φ(1 R )=1 S <br />
R φ(R) ≠ {0} φ(R) <br />
<br />
<br />
<br />
R I R a I r R<br />
ar ra I rI ⊂ I Ir ⊂ I r ∈ R<br />
R {0} R <br />
<br />
R I R 1 I<br />
r ∈ R r1 =r ∈ I I = R<br />
a R <br />
〈a〉 = {ar : r ∈ R}<br />
R 〈a〉 0=a0 a = a1 〈a〉 <br />
〈a〉 〈a〉 ar + ar ′ = a(r + r ′ ) ar <br />
−ar = a(−r) ∈〈a〉 ar ∈〈a〉 <br />
s ∈ R s(ar) =a(sr) 〈a〉 <br />
R 〈a〉 = {ar : r ∈ R}<br />
<br />
Z <br />
{0} 〈0〉 = {0} I <br />
Z I m n<br />
I a I <br />
q r <br />
a = nq + r<br />
0 ≤ r
I ⊂ I Ir ⊂ I <br />
r ∈ R <br />
rI ⊂ I Ir ⊂ I r ∈ R<br />
<br />
<br />
<br />
I R R/I <br />
<br />
(r + I)(s + I) =rs + I.<br />
R/I r + I <br />
s + I R/I (r + I)(s + I) =rs + I <br />
r ′ ∈ r + I s ′ ∈ s + I r ′ s ′ rs + I <br />
r ′ ∈ r + I a I r ′ = r + a b ∈ I<br />
s ′ = s + b <br />
r ′ s ′ =(r + a)(s + b) =rs + as + rb + ab<br />
as + rb + ab ∈ I I r ′ s ′ ∈ rs + I <br />
<br />
<br />
R/I <br />
<br />
<br />
I R φ : R → R/I φ(r) =r + I<br />
R R/I I<br />
φ : R → R/I <br />
φ r s R <br />
<br />
φ(r)φ(s) =(r + I)(s + I) =rs + I = φ(rs),<br />
φ : R → R/I <br />
<br />
<br />
<br />
<br />
<br />
ψ : R → S <br />
ψ R φ : R → R/ ψ <br />
η : R/ ψ → ψ(R) ψ = ηφ<br />
K = ψ <br />
η : R/K → ψ(R) η(r + K) =ψ(r) <br />
R R/K <br />
η((r + K)(s + K)) = η(r + K)η(s + K) <br />
η((r + K)(s + K)) = η(rs + K)<br />
= ψ(rs)<br />
= ψ(r)ψ(s)<br />
= η(r + K)η(s + K).
I R <br />
J R I ∩ J I <br />
I/I ∩ J ∼ = (I + J)/J.<br />
R I J <br />
R J ⊂ I <br />
R/I ∼ = R/J<br />
I/J .<br />
I R <br />
S ↦→ S/I S I <br />
R/I R I <br />
R/I<br />
<br />
<br />
<br />
<br />
I R R/I <br />
<br />
M R R M <br />
R R M I<br />
M I = R <br />
<br />
R M R <br />
M R R/M <br />
M R R R/M <br />
1+M R/M <br />
R/M a + M R/M<br />
a /∈ M I {ra + m : r ∈ R m ∈ M} I <br />
R I 0a +0=0 I r 1 a + m 1 r 2 a + m 2 <br />
I <br />
(r 1 a + m 1 ) − (r 2 a + m 2 )=(r 1 − r 2 )a +(m 1 − m 2 )<br />
I r ∈ R rI ⊂ I I <br />
<br />
I M M <br />
I = R I m M b <br />
R 1=ab + m <br />
1+M = ab + M = ba + M =(a + M)(b + M).<br />
M R/M R/M <br />
0+M = M 1+M M <br />
R I M I = R <br />
a I M a + M <br />
b + M R/M (a + M)(b + M) =ab + M =1+M <br />
m ∈ M ab + m =1 1 I r1 =r ∈ I r ∈ R<br />
I = R
pZ Z p pZ <br />
Z/pZ ∼ = Z p <br />
P R ab ∈ P <br />
a ∈ P b ∈ P <br />
P = {0, 2, 4, 6, 8, 10} Z 12 <br />
<br />
R 1 1 ≠0 P <br />
R R/P <br />
P R R/P <br />
ab ∈ P a+P b+P R/P (a+P )(b+P )=0+P = P <br />
a + P = P b + P = P a P b P <br />
P <br />
P <br />
(a + P )(b + P )=ab + P =0+P = P.<br />
ab ∈ P a /∈ P b P <br />
b + P =0+P R/P <br />
Z nZ Z/nZ ∼ = Z n <br />
n <br />
Z pZ p
m n (m, n) =1 a, b ∈ Z<br />
<br />
x ≡ a ( m)<br />
x ≡ b ( n)<br />
x 1 x 2 x 1 ≡ x 2 ( mn)<br />
x ≡ a ( m) a + km <br />
k ∈ Z k 1 <br />
a + k 1 m ≡ b<br />
( n).<br />
<br />
k 1 m ≡ (b − a) ( n)<br />
k 1 m n s t <br />
ms + nt =1 <br />
(b − a)ms =(b − a) − (b − a)nt,<br />
<br />
[(b − a)s]m ≡ (b − a) ( n).<br />
k 1 =(b − a)s<br />
mn c 1 c 2 <br />
<br />
c i ≡ a ( m)
i =1, 2 <br />
c i ≡ b ( n)<br />
c 2 ≡ c 1 ( m)<br />
c 2 ≡ c 1 ( n).<br />
m n c 1 − c 2 c 2 ≡ c 1 ( mn)<br />
<br />
x ≡ 3 ( 4)<br />
x ≡ 4 ( 5).<br />
s t 4s +5t =1 <br />
s =4 t = −3 <br />
x = a + k 1 m =3+4k 1 =3+4[(5− 4)4] = 19.<br />
n 1 ,n 2 ,...,n k <br />
(n i ,n j )=1 i ≠ j a 1 ,...,a k <br />
x ≡ a 1 ( n 1 )<br />
x ≡ a 2 ( n 2 )<br />
<br />
x ≡ a k ( n k )<br />
<br />
n 1 n 2 ···n k <br />
<br />
<br />
k =2 <br />
k <br />
x ≡ a 1 ( n 1 )<br />
x ≡ a 2 ( n 2 )<br />
<br />
x ≡ a k+1 ( n k+1 ).<br />
k n 1 ···n k <br />
a n 1 ···n k n k+1 <br />
x ≡ a ( n 1 ···n k )<br />
x ≡ a k+1 ( n k+1 )<br />
n 1 ...n k+1 <br />
<br />
x ≡ 3 ( 4)<br />
x ≡ 4 ( 5)<br />
x ≡ 1 ( 9)
x ≡ 5 ( 7).<br />
19 <br />
19 ( 20) <br />
<br />
x ≡ 19 ( 20)<br />
x ≡ 1 ( 9)<br />
x ≡ 5 ( 7).<br />
<br />
x ≡ 19 ( 180)<br />
x ≡ 5 ( 7).<br />
19 <br />
1260<br />
<br />
<br />
<br />
<br />
<br />
<br />
2 63 − 1=9,223,372,036,854,775,807.<br />
<br />
<br />
2 511 − 1 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
2134 1531 <br />
95 97 98 99 <br />
<br />
2134 ≡ 44 ( 95)<br />
2134 ≡ 0 ( 97)<br />
2134 ≡ 76 ( 98)<br />
2134 ≡ 55 ( 99)
1531 ≡ 11 ( 95)<br />
1531 ≡ 76 ( 97)<br />
1531 ≡ 61 ( 98)<br />
1531 ≡ 46 ( 99).<br />
<br />
2134 · 1531 ≡ 44 · 11 ≡ 9 ( 95)<br />
2134 · 1531 ≡ 0 · 76 ≡ 0 ( 97)<br />
2134 · 1531 ≡ 76 · 61 ≡ 30 ( 98)<br />
2134 · 1531 ≡ 55 · 46 ≡ 55 ( 99).<br />
<br />
2134·1531 <br />
x ≡ 9 ( 95)<br />
x ≡ 0 ( 97)<br />
x ≡ 30 ( 98)<br />
x ≡ 55 ( 99).<br />
95 · 97 ·<br />
98 · 99 = 89,403,930 x 2134 · 1531 =<br />
3,267,154<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
7Z<br />
Z 18<br />
Q( √ 2)={a + b √ 2:a, b ∈ Q}<br />
Q( √ 2, √ 3)={a + b √ 2+c √ 3+d √ 6:a, b, c, d ∈ Q}<br />
Z[ √ 3]={a + b √ 3:a, b ∈ Z}<br />
R = {a + b 3√ 3:a, b ∈ Q}<br />
Z[i] ={a + bi : a, b ∈ Z i 2 = −1}
Q( 3√ 3)={a + b 3√ 3+c 3√ 9:a, b, c ∈ Q}<br />
R 2 × 2 <br />
( ) a b<br />
,<br />
0 0<br />
a, b ∈ R R <br />
S R <br />
<br />
Z 10<br />
Z 12<br />
Z 7<br />
M 2 (Z) 2 × 2 Z<br />
M 2 (Z 2 ) 2 × 2 Z 2<br />
<br />
<br />
Z 18<br />
Z 25<br />
M 2 (R) 2 × 2 R<br />
M 2 (Z) 2 × 2 Z<br />
Q<br />
R I <br />
R/I<br />
R = Z I =6Z<br />
R = Z 12 I = {0, 3, 6, 9}<br />
φ : Z/6Z → Z/15Z<br />
R C<br />
Q( √ 2) = {a + b √ 2:a, b ∈ Q} <br />
Q( √ 3)={a + b √ 3:a, b ∈ Q}<br />
<br />
{( ) ( ) ( ) ( )}<br />
1 0 1 1 0 1 0 0<br />
F = , , ,<br />
0 1 1 0 1 1 0 0<br />
Z 2 <br />
φ : C → M 2 (R) <br />
( ) a b<br />
φ(a + bi) = .<br />
−b a<br />
φ C M 2 (R)<br />
Z[i] <br />
Z[ √ 3 i] ={a + b √ 3 i : a, b ∈ Z}
x ≡ 2 ( 5)<br />
x ≡ 6 ( 11)<br />
x ≡ 4 ( 7)<br />
x ≡ 7 ( 9)<br />
x ≡ 5 ( 11)<br />
<br />
<br />
x ≡ 3 ( 7)<br />
x ≡ 0 ( 8)<br />
x ≡ 5 ( 15)<br />
x ≡ 2 ( 4)<br />
<br />
x ≡ 3 ( 5)<br />
x ≡ 0 ( 8)<br />
x ≡ 1 ( 11)<br />
x ≡ 5 ( 13)<br />
2234 + 4121<br />
95 97 98 99<br />
2134 ·<br />
1531 98<br />
99<br />
R R {0} R <br />
a R (−1)a = −a<br />
φ : R → S <br />
R φ(R) <br />
φ(0) = 0<br />
1 R 1 S R S φ φ(1 R )=1 S <br />
R φ(R) ≠0 φ(R) <br />
<br />
R/I<br />
I R<br />
J R I ∩ J I <br />
I/I ∩ J ∼ = I + J/J.<br />
R I J <br />
R J ⊂ I <br />
R/I ∼ = R/J<br />
I/J .<br />
I R S → S/I<br />
S I <br />
R/I R R/I<br />
R S R S R <br />
<br />
S ≠ ∅<br />
rs ∈ S r, s ∈ S
− s ∈ S r, s ∈ S<br />
R {R α } ⋂ R α <br />
R <br />
{I α } α∈A R ⋂ α∈A I α <br />
R I 1 I 2 R I 1 ∪ I 2 <br />
<br />
R R {0} R <br />
R <br />
R a R a n =0 <br />
n R<br />
R a ∈ R a 2 = a <br />
<br />
R a 3 = a a ∈ R R <br />
<br />
R 1 R S R 1 S <br />
1 R =1 S <br />
<br />
R 1=0 <br />
R = {0}<br />
S R R ′ R <br />
S<br />
R R <br />
Z(R) ={a ∈ R : ar = ra r ∈ R}.<br />
Z(R) R<br />
p <br />
Z (p) = {a/b : a, b ∈ Z (b, p) =1}<br />
Z (p) p<br />
Z p <br />
R <br />
u R i u : R → R r ↦→ uru −1 i u <br />
R R <br />
R R (R)<br />
R (R) (R) <br />
(R)<br />
U(R) R <br />
φ : U(R) → (R)<br />
u ↦→ i u φ<br />
(Z) (Z) U(Z)<br />
R S <br />
R × S
(r, s)+(r ′ ,s ′ )=(r + r ′ ,s+ s ′ )<br />
(r, s)(r ′ ,s ′ )=(rr ′ ,ss ′ )<br />
x x 2 = x <br />
0 1 x <br />
<br />
(a, n) =d (b, d) ≠1 ax ≡ b ( n) <br />
<br />
R I J <br />
R I + J = R<br />
r s R <br />
<br />
x ≡ r ( I)<br />
x ≡ s ( J)<br />
I ∩ J<br />
I J R I + J = R <br />
<br />
R/(I ∩ J) ∼ = R/I × R/J.
17<br />
<br />
<br />
<br />
p(x) =x 3 − 3x +2<br />
q(x) =3x 2 − 6x +5,<br />
p(x) +q(x) p(x)q(x) <br />
<br />
<br />
(p + q)(x) =p(x)+q(x)<br />
=(x 3 − 3x +2)+(3x 2 − 6x +5)<br />
= x 3 +3x 2 − 9x +7<br />
<br />
(pq)(x) =p(x)q(x)<br />
=(x 3 − 3x + 2)(3x 2 − 6x +5)<br />
=3x 5 − 6x 4 − 4x 3 +24x 2 − 27x +10.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
R <br />
<br />
f(x) =<br />
n∑<br />
a i x i = a 0 + a 1 x + a 2 x 2 + ···+ a n x n ,<br />
i=0<br />
a i ∈ R a n ≠0 R x <br />
a 0 ,a 1 ,...,a n f a n <br />
n <br />
a n ≠0 f n <br />
f(x) =n n f =0
f −∞ <br />
R R[x] <br />
<br />
p(x) =a 0 + a 1 x + ···+ a n x n<br />
q(x) =b 0 + b 1 x + ···+ b m x m ,<br />
p(x) =q(x) a i = b i i ≥ 0<br />
<br />
<br />
p(x) q(x) <br />
p(x) =a 0 + a 1 x + ···+ a n x n<br />
q(x) =b 0 + b 1 x + ···+ b m x m .<br />
p(x)+q(x) =c 0 + c 1 x + ···+ c k x k ,<br />
c i = a i + b i i p(x) q(x) <br />
<br />
c i =<br />
p(x)q(x) =c 0 + c 1 x + ···+ c m+n x m+n ,<br />
i∑<br />
a k b i−k = a 0 b i + a 1 b i−1 + ···+ a i−1 b 1 + a i b 0<br />
k=0<br />
i <br />
<br />
p(x) =3+0x +0x 2 +2x 3 +0x 4<br />
<br />
q(x) =2+0x − x 2 +0x 3 +4x 4<br />
Z[x] <br />
p(x) = 3 + 2x 3 q(x) =<br />
2 − x 2 +4x 4 <br />
<br />
p(x)+q(x) =5− x 2 +2x 3 +4x 4 .<br />
p(x)q(x) =(3+2x 3 )(2 − x 2 +4x 4 )=6− 3x 2 +4x 3 +12x 4 − 2x 5 +8x 7 ,<br />
c i <br />
<br />
<br />
p(x) =3+3x 3 q(x) =4+4x 2 +4x 4<br />
Z 12 [x] p(x) q(x) 7+4x 2 +3x 3 +4x 4 <br />
<br />
R[x] R
R R[x] <br />
<br />
R[x] <br />
∑<br />
f(x) =0 p(x) =<br />
n<br />
i=0 a ix i p(x) −p(x) = ∑ n<br />
i=0 (−a i)x i = − ∑ n<br />
i=0 a ix i <br />
<br />
R <br />
<br />
p(x) =<br />
q(x) =<br />
r(x) =<br />
m∑<br />
a i x i ,<br />
i=0<br />
n∑<br />
b i x i ,<br />
i=0<br />
p∑<br />
c i x i .<br />
i=0<br />
<br />
[( m<br />
)(<br />
∑<br />
n∑<br />
)] ( p∑<br />
)<br />
[p(x)q(x)]r(x) = a i x i b i x i c i x i<br />
i=0 i=0<br />
i=0<br />
⎡ ⎛ ⎞ ⎤ )<br />
m+n<br />
∑ i∑<br />
= ⎣ ⎝ a j b i−j<br />
⎠ x i ⎦ c i x i<br />
=<br />
=<br />
=<br />
i=0<br />
m+n+p<br />
∑<br />
i=0<br />
m+n+p<br />
∑<br />
i=0<br />
m+n+p<br />
∑<br />
i=0<br />
j=0<br />
j+k+l=i<br />
( p∑<br />
i=0<br />
⎡ (<br />
i∑ j∑<br />
)<br />
⎣ a k b j−k<br />
j=0 k=0<br />
⎛<br />
⎞<br />
⎝<br />
∑<br />
a j b k c l<br />
⎠ x i<br />
⎡<br />
⎣<br />
i∑<br />
j=0<br />
( m<br />
) ⎡ ∑<br />
= a i x i ⎣<br />
=<br />
i=0<br />
( m<br />
∑<br />
c i−j<br />
⎤<br />
⎦ x i<br />
( i−j<br />
) ⎤ ∑<br />
a j b k c i−j−k<br />
⎦ x i<br />
k=0<br />
⎛ ⎞ ⎤<br />
n+p<br />
∑ i∑<br />
⎝ b j c i−j<br />
⎠ x i ⎦<br />
i=0<br />
)[( n∑<br />
a i x i<br />
i=0 i=0<br />
= p(x)[q(x)r(x)]<br />
j=0<br />
b i x i )( p∑<br />
i=0<br />
c i x i )]<br />
<br />
<br />
p(x) q(x) R[x] R <br />
p(x)+ q(x) =(p(x)q(x)) R[x] <br />
<br />
<br />
p(x) =a m x m + ···+ a 1 x + a 0
q(x) =b n x n + ···+ b 1 x + b 0<br />
a m ≠0 b n ≠0 p(x) q(x) m n <br />
p(x)q(x) a m b n x m+n R <br />
p(x)q(x) m + n p(x)q(x) ≠0 p(x) ≠0 q(x) ≠0<br />
p(x)q(x) ≠0 R[x] <br />
x 2 − 3xy +2y 3 <br />
R x y <br />
(R[x])[y] <br />
(R[x])[y] ∼ = R([y])[x] <br />
R[x, y] R[x, y] x <br />
y R n <br />
R R[x 1 ,x 2 ,...,x n ]<br />
R α ∈ R <br />
φ α : R[x] → R <br />
p(x) =a n x n + ···+ a 1 x + a 0 <br />
φ α (p(x)) = p(α) =a n α n + ···+ a 1 α + a 0 ,<br />
p(x) = ∑ n<br />
i=0 a ix i q(x) = ∑ m<br />
i=0 b ix i φ α (p(x) +<br />
q(x)) = φ α (p(x)) + φ α (q(x)) φ α <br />
<br />
φ α (p(x))φ α (q(x)) = p(α)q(α)<br />
( n∑<br />
)( m<br />
)<br />
∑<br />
= a i α i b i α i<br />
i=0<br />
i=0<br />
( i∑<br />
)<br />
a k b i−k<br />
k=0<br />
=<br />
m+n<br />
∑<br />
i=0<br />
= φ α (p(x)q(x)).<br />
φ α : R[x] → R α<br />
α i<br />
<br />
<br />
a b <br />
b>0 q r a = bq + r 0 ≤ r
q(x) r(x) f(x) <br />
<br />
0=0· g(x)+0;<br />
q r f(x) <br />
f(x) =n g(x) =m m>n <br />
q(x) =0 r(x) =f(x) m ≤ n <br />
n <br />
f(x) =a n x n + a n−1 x n−1 + ···+ a 1 x + a 0<br />
g(x) =b m x m + b m−1 x m−1 + ···+ b 1 x + b 0<br />
<br />
f ′ (x) =f(x) − a n<br />
x n−m g(x)<br />
b m<br />
n q ′ (x)<br />
r(x) <br />
f ′ (x) =q ′ (x)g(x)+r(x),<br />
r(x) =0 r(x) g(x) <br />
q(x) =q ′ (x)+ a n<br />
x n−m .<br />
b m<br />
<br />
f(x) =g(x)q(x)+r(x),<br />
r(x) r(x) < g(x)<br />
q(x) r(x) <br />
q 1 (x) r 1 (x) f(x) =g(x)q 1 (x)+r 1 (x) r 1 (x) < g(x) r 1 (x) =0<br />
<br />
f(x) =g(x)q(x)+r(x) =g(x)q 1 (x)+r 1 (x),<br />
<br />
g(x)[q(x) − q 1 (x)] = r 1 (x) − r(x).<br />
q(x) − q 1 (x) <br />
(g(x)[q(x) − q 1 (x)]) = (r 1 (x) − r(x)) ≥ g(x).<br />
r(x) r 1 (x) g(x)<br />
r(x) =r 1 (x) q(x) =q 1 (x)<br />
<br />
<br />
x 3 − x 2 +2x − 3 x − 2<br />
x 2 + x + 4<br />
x − 2 x 3 − x 2 + 2x − 3<br />
x 3 − 2x 2<br />
x 2 + 2x − 3<br />
x 2 − 2x<br />
4x − 3<br />
4x − 8<br />
5<br />
x 3 − x 2 +2x − 3=(x − 2)(x 2 + x +4)+5
p(x) F [x] α ∈ F α p(x) <br />
p(x) φ α <br />
α p(x) p(α) =0<br />
F α ∈ F p(x) ∈ F [x] <br />
x − α p(x) F [x]<br />
α ∈ F p(α) = 0 <br />
q(x) r(x) <br />
p(x) =(x − α)q(x)+r(x)<br />
r(x) x − α r(x) <br />
r(x) =a a ∈ F <br />
p(x) =(x − α)q(x)+a.<br />
<br />
0=p(α) =0· q(α)+a = a;<br />
p(x) =(x − α)q(x) x − α p(x)<br />
x − α p(x) p(x) = (x − α)q(x)<br />
p(α) =0· q(α) =0<br />
<br />
F p(x) n F [x] <br />
n F <br />
p(x) p(x) =0 p(x) <br />
p(x) =1 p(x) =ax + b a<br />
b F α 1 α 2 p(x) aα 1 + b = aα 2 + b α 1 = α 2 <br />
p(x) > 1 p(x) F <br />
α p(x) p(x) =(x − α)q(x) q(x) ∈ F [x] <br />
q(x) n − 1 β <br />
p(x) α p(β) =(β − α)q(β) =0 α ≠ β F <br />
q(β) =0 q(x) n − 1 F <br />
α p(x) n F <br />
F d(x) <br />
p(x),q(x) ∈ F [x] d(x) p(x) q(x) <br />
d ′ (x) p(x) q(x) d ′ (x) | d(x) d(x) =(p(x),q(x))<br />
p(x) q(x) (p(x),q(x)) = 1<br />
F d(x) <br />
p(x) q(x) F [x] r(x) s(x) <br />
<br />
d(x) =r(x)p(x)+s(x)q(x).<br />
<br />
<br />
d(x) <br />
S = {f(x)p(x)+g(x)q(x) :f(x),g(x) ∈ F [x]}.<br />
d(x) =r(x)p(x) +s(x)q(x) r(x) s(x) F [x] <br />
d(x) p(x) q(x) d(x)
p(x) a(x) b(x) p(x) =<br />
a(x)d(x)+b(x) b(x) b(x) < d(x) <br />
b(x) =p(x) − a(x)d(x)<br />
= p(x) − a(x)(r(x)p(x)+s(x)q(x))<br />
= p(x) − a(x)r(x)p(x) − a(x)s(x)q(x)<br />
= p(x)(1 − a(x)r(x)) + q(x)(−a(x)s(x))<br />
p(x) q(x) S b(x) <br />
d(x) d(x)<br />
p(x) d(x) q(x) d(x) <br />
p(x) q(x)<br />
d(x) p(x) q(x) d ′ (x)<br />
p(x) q(x) d ′ (x) | d(x) d ′ (x)<br />
p(x) q(x) u(x) v(x) <br />
p(x) =u(x)d ′ (x) q(x) =v(x)d ′ (x) <br />
d(x) =r(x)p(x)+s(x)q(x)<br />
= r(x)u(x)d ′ (x)+s(x)v(x)d ′ (x)<br />
= d ′ (x)[r(x)u(x)+s(x)v(x)].<br />
d ′ (x) | d(x) d(x) p(x) q(x)<br />
p(x) q(x) <br />
d ′ (x) p(x) q(x) <br />
u(x) v(x) F [x] d(x) =d ′ (x)[r(x)u(x)+<br />
s(x)v(x)] <br />
d(x) = d ′ (x)+[r(x)u(x)+s(x)v(x)]<br />
d(x) d ′ (x) d(x) = d ′ (x) d(x)<br />
d ′ (x) d(x) =<br />
d ′ (x)<br />
<br />
<br />
<br />
<br />
f(x) ∈ F [x] F f(x) <br />
g(x) h(x) F [x] g(x)<br />
h(x) f(x) <br />
<br />
x 2 − 2 ∈ Q[x] <br />
x 2 +1 <br />
p(x) =x 3 + x 2 +2 Z 3 [x] <br />
Z 3 [x] <br />
x − a a Z 3 [x] <br />
p(a) =0 <br />
p(0) = 2<br />
p(1) = 1
p(2) = 2.<br />
p(x) Z 3 <br />
p(x) ∈ Q[x] <br />
p(x) = r s (a 0 + a 1 x + ···+ a n x n ),<br />
r, s, a 0 ,...,a n a i r s <br />
<br />
<br />
p(x) = b 0<br />
+ b 1<br />
x + ···+ b n<br />
x n ,<br />
c 0 c 1 c n<br />
b i c i p(x) <br />
1<br />
p(x) = (d 0 + d 1 x + ···+ d n x n ),<br />
c 0 ···c n<br />
d 0 ,...,d n d d 0 ,...,d n <br />
d<br />
p(x) = (a 0 + a 1 x + ···+ a n x n ),<br />
c 0 ···c n<br />
d i = da i a i d/(c 0 ···c n ) <br />
<br />
p(x) = r s (a 0 + a 1 x + ···+ a n x n ),<br />
(r, s) =1<br />
p(x) ∈ Z[x] p(x)<br />
α(x) β(x) Q[x] <br />
α(x) β(x) p(x) p(x) =a(x)b(x) a(x) b(x)<br />
Z[x] α(x) = a(x) β(x) = b(x)<br />
<br />
<br />
α(x) = c 1<br />
(a 0 + a 1 x + ···+ a m x m )= c 1<br />
α 1 (x)<br />
d 1 d 1<br />
β(x) = c 2<br />
(b 0 + b 1 x + ···+ b n x n )= c 2<br />
β 1 (x),<br />
d 2 d 2<br />
a i b i <br />
p(x) =α(x)β(x) = c 1c 2<br />
d 1 d 2<br />
α 1 (x)β 1 (x) = c d α 1(x)β 1 (x),<br />
c/d c 1 /d 1 c 2 /d 2 dp(x) =<br />
cα 1 (x)β 1 (x)<br />
d = 1 ca m b n = 1 p(x) c = 1<br />
c = −1 c = 1 a m = b n = 1 a m = b n = −1 <br />
p(x) =α 1 (x)β 1 (x) α 1 (x) β 1 (x) α(x) = α 1 (x)<br />
β(x) = β 1 (x) a(x) =−α 1 (x) b(x) =−β 1 (x) <br />
p(x) =(−α 1 (x))(−β 1 (x)) = a(x)b(x) <br />
c = −1 <br />
d ≠1 (c, d) =1 p p | d <br />
p ∤ c α 1 (x) a i
p ∤ a i b j β 1 (x) p ∤ b j α ′ 1 (x)<br />
β ′ 1 (x) Z p[x] α 1 (x) <br />
β 1 (x) p p | d α ′ 1 (x)β′ 1 (x) =0 Z p[x] <br />
α ′ 1 (x) β′ 1 (x) Z p[x] <br />
d =1 <br />
p(x) =x n + a n−1 x n−1 + ···+ a 0 <br />
Z a 0 ≠0 p(x) Q p(x) α Z α<br />
a 0 <br />
p(x) a ∈ Q p(x) x − a <br />
p(x) Z[x] α ∈ Z<br />
a 0 /α ∈ Z α | a 0 <br />
p(x) =(x − α)(x n−1 + ···−a 0 /α).<br />
p(x) =x 4 − 2x 3 + x +1 p(x) <br />
Q[x] p(x) p(x) p(x) =<br />
(x − α)q(x) q(x) p(x) <br />
p(x) Q[x] Z <br />
±1 p(1) = 1 p(−1) = 3 <br />
p(x) <br />
p(x) <br />
p(x) =(x 2 + ax + b)(x 2 + cx + d)<br />
= x 4 +(a + c)x 3 +(ac + b + d)x 2 +(ad + bc)x + bd,<br />
Z[x] <br />
a + c = −2<br />
ac + b + d =0<br />
ad + bc =1<br />
bd =1.<br />
bd =1 b = d =1 b = d = −1 b = d <br />
ad + bc = b(a + c) =1.<br />
a+c = −2 −2b =1 b <br />
p(x) Q<br />
p <br />
f(x) =a n x n + ···+ a 0 ∈ Z[x].<br />
p | a i i =0, 1,...,n− 1 p ∤ a n p 2 ∤ a 0 f(x) Q<br />
f(x) <br />
Z[x] <br />
f(x) =(b r x r + ···+ b 0 )(c s x s + ···+ c 0 )<br />
Z[x] b r c s r,s
p ∤ a n a n = b r c s b r c s p m k<br />
p ∤ c k <br />
a m = b 0 c m + b 1 c m−1 + ···+ b m c 0<br />
p p<br />
b 0 c m m = n a i p m
〈p(x)〉 p(x) 〈p(x)〉 ⊂〈f(x)〉<br />
〈p(x)〉 <br />
p(x) F [x] I F [x] <br />
〈p(x)〉 I I = 〈f(x)〉 f(x) ∈ F [x]<br />
p(x) ∈ I p(x) =f(x)g(x) g(x) ∈ F [x] <br />
p(x) f(x) g(x) f(x) <br />
I = F [x] g(x) f(x) I<br />
I = 〈p(x)〉 F [x] 〈p(x)〉<br />
<br />
<br />
<br />
<br />
<br />
ax 2 + bx + c =0 <br />
<br />
<br />
ax 3 + bx 2 + cx + d =0 <br />
<br />
<br />
<br />
<br />
<br />
ax 3 + cx + d =0.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
ax 3 + bx 2 + cx + d =0.
ax 4 + bx 3 + cx 2 + dx + e =0.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
3 Z 2 [x]<br />
<br />
(5x 2 +3x − 4) + (4x 2 − x +9) Z 12<br />
(5x 2 +3x − 4)(4x 2 − x +9) Z 12<br />
(7x 3 +3x 2 − x)+(6x 2 − 8x +4) Z 9<br />
(3x 2 +2x − 4) + (4x 2 +2) Z 5<br />
(3x 2 +2x − 4)(4x 2 +2) Z 5<br />
(5x 2 +3x − 2) 2 Z 12<br />
q(x) r(x) a(x) =q(x)b(x)+r(x) <br />
r(x) < b(x) <br />
a(x) =5x 3 +6x 2 − 3x +4 b(x) =x − 2 Z 7 [x]<br />
a(x) =6x 4 − 2x 3 + x 2 − 3x +1 b(x) =x 2 + x − 2 Z 7 [x]<br />
a(x) =4x 5 − x 3 + x 2 +4 b(x) =x 3 − 2 Z 5 [x]<br />
a(x) =x 5 + x 3 − x 2 − x b(x) =x 3 + x Z 2 [x]<br />
p(x) q(x) <br />
d(x) = (p(x),q(x)) a(x) b(x) <br />
a(x)p(x)+b(x)q(x) =d(x)<br />
p(x) =x 3 − 6x 2 +14x − 15 q(x) =x 3 − 8x 2 +21x − 18 p(x),q(x) ∈ Q[x]<br />
p(x) =x 3 + x 2 − x +1 q(x) =x 3 + x − 1 p(x),q(x) ∈ Z 2 [x]<br />
p(x) =x 3 + x 2 − 4x +4 q(x) =x 3 +3x − 2 p(x),q(x) ∈ Z 5 [x]<br />
p(x) =x 3 − 2x +4 q(x) =4x 3 + x +3 p(x),q(x) ∈ Q[x]<br />
<br />
<br />
5x 3 +4x 2 − x +9 Z 12<br />
3x 3 − 4x 2 − x +4 Z 5<br />
5x 4 +2x 2 − 3 Z 7<br />
x 3 + x +1 Z 2
Z[x]<br />
p(x) Z 4 [x] p(x) > 1<br />
Q[x]<br />
x 4 − 2x 3 +2x 2 + x +4<br />
x 4 − 5x 3 +3x − 2<br />
3x 5 − 4x 3 − 6x 2 +6<br />
5x 5 − 6x 4 − 3x 2 +9x − 15<br />
2 3 Z 2 [x]<br />
x 2 + x +8 Z 10 [x]<br />
p(x) Z 6 [x] n <br />
n <br />
F F [x 1 ,...,x n ] <br />
Z[x] <br />
x p + a a ∈ Z p p <br />
f(x) F [x] F f(x) | p(x)q(x) <br />
f(x) | p(x) f(x) | q(x)<br />
R S R[x] ∼ = S[x]<br />
F a ∈ F p(x) ∈ F [x] p(a) <br />
p(x) x − a<br />
<br />
p(x) =a n x n + a n−1 x n−1 + ···+ a 0 ∈ Z[x],<br />
a n ≠0 p(r/s) =0 (r, s) =1 r | a 0 s | a n <br />
Q ∗ Q ∗ <br />
(Z[x], +)<br />
<br />
Φ n (x) = xn − 1<br />
x − 1 = xn−1 + x n−2 + ···+ x +1<br />
Φ p (x) Q <br />
p<br />
F F [x]<br />
R <br />
R[x]<br />
R <br />
R[x]<br />
x p − x p Z p p <br />
x p − x = x(x − 1)(x − 2) ···(x − (p − 1)).<br />
F f(x) =a 0 + a 1 x + ··· + a n x n F [x] f ′ (x) =<br />
a 1 +2a 2 x + ···+ na n x n−1 f(x)<br />
<br />
(f + g) ′ (x) =f ′ (x)+g ′ (x).<br />
D : F [x] → F [x] <br />
D(f(x)) = f ′ (x)
D F =0<br />
D F = p<br />
<br />
(fg) ′ (x) =f ′ (x)g(x)+f(x)g ′ (x).<br />
f(x) ∈ F [x] <br />
f(x) =a(x − a 1 )(x − a 2 ) ···(x − a n ).<br />
f(x) f(x) f ′ (x) <br />
<br />
F F [x] <br />
R R[x 1 ,...,x n ] <br />
R R[x] R ′<br />
R<br />
p(x) q(x) R[x] R <br />
(p(x)+q(x)) ≤ ( p(x), q(x))<br />
<br />
<br />
<br />
<br />
ax 2 + bx + c =0<br />
<br />
x = −b ± √ b 2 − 4ac<br />
.<br />
2a<br />
Δ=b 2 − 4ac <br />
Δ > 0 Δ=0<br />
Δ < 0 <br />
<br />
<br />
x 3 + bx 2 + cx + d =0<br />
y 3 + py + q =0 x = y − b/3<br />
<br />
ω = −1+i√ 3<br />
2<br />
ω 2 = −1 − i√ 3<br />
2<br />
ω 3 =1.<br />
<br />
<br />
y = z − p<br />
3z
y y 3 + py + q =0 A B z 3 <br />
−p 3 /27 <br />
3√<br />
AB = −p/3<br />
<br />
z <br />
3√<br />
A, ω<br />
3 √ A, ω 2 3 √ A,<br />
3√<br />
B, ω<br />
3 √ B, ω 2 3 √ B<br />
y <br />
√<br />
ω i 3 − q √<br />
√<br />
p<br />
2 + 3<br />
27 + q2<br />
4 + ω2i 3 − q √<br />
p<br />
2 − 3<br />
27 + q2<br />
4 ,<br />
i =0, 1, 2<br />
<br />
y 3 + py + q =0<br />
Δ= p3<br />
27 + q2<br />
4 .<br />
Δ=0<br />
Δ > 0<br />
Δ < 0<br />
<br />
<br />
x 3 − 4x 2 +11x +30=0<br />
x 3 − 3x +5=0<br />
x 3 − 3x +2=0<br />
x 3 + x +3=0<br />
<br />
<br />
<br />
x 4 + ax 3 + bx 2 + cx + d =0<br />
y 4 + py 2 + qy + r =0<br />
x = y − a/4<br />
(<br />
y 2 + 1 ) 2 ( ) 1<br />
2 z =(z − p)y 2 − qy +<br />
4 z2 − r .<br />
(my + k) 2<br />
<br />
( ) 1<br />
q 2 − 4(z − p)<br />
4 z2 − r =0.<br />
<br />
<br />
z 3 − pz 2 − 4rz +(4pr − q 2 )=0.
(<br />
y 2 + 1 ) 2<br />
2 z =(my + k) 2<br />
<br />
<br />
x 4 − x 2 − 3x +2=0<br />
x 4 + x 3 − 7x 2 − x +6=0<br />
x 4 − 2x 2 +4x − 3=0<br />
x 4 − 4x 3 +3x 2 − 5x +2=0
18<br />
<br />
<br />
Z[x] <br />
Q <br />
<br />
<br />
<br />
<br />
<br />
<br />
Z <br />
<br />
Q <br />
<br />
<br />
D F <br />
D <br />
p/q ∈ Q p q <br />
1/2 = 2/4 = 3/6 <br />
<br />
a<br />
b = c d<br />
ad = bc <br />
Q Z × Z<br />
p/q (p, q) (3, 7) <br />
3/7 Z × Z <br />
5/0 (5, 0) (3, 6) (2, 4) <br />
1/2 <br />
(a, b) (c, d) <br />
ad = bc<br />
<br />
D <br />
S = {(a, b) :a, b ∈ D b ≠0}.<br />
S (a, b) ∼ (c, d) ad = bc<br />
∼ S
D ab = ba ∼ D <br />
(a, b) ∼ (c, d) ad = bc cb = da (c, d) ∼ (a, b) <br />
(a, b) ∼ (c, d) (c, d) ∼<br />
(e, f) ad = bc cf = de ad = bc f <br />
afd = adf = bcf = bde = bed.<br />
D af = be (a, b) ∼ (e, f)<br />
S F D <br />
F D <br />
Q<br />
a<br />
b + c ad + bc<br />
= ;<br />
d bd<br />
a<br />
b · c<br />
d = ac<br />
bd .<br />
F D <br />
(a, b) ∈ S [a, b] <br />
F D <br />
<br />
[a, b]+[c, d] =[ad + bc, bd]<br />
[a, b] · [c, d] =[ac, bd],<br />
<br />
<br />
F D <br />
<br />
[a 1 ,b 1 ]=[a 2 ,b 2 ] [c 1 ,d 1 ]=[c 2 ,d 2 ]<br />
<br />
[a 1 d 1 + b 1 c 1 ,b 1 d 1 ]=[a 2 d 2 + b 2 c 2 ,b 2 d 2 ]<br />
<br />
(a 1 d 1 + b 1 c 1 )(b 2 d 2 )=(b 1 d 1 )(a 2 d 2 + b 2 c 2 ).<br />
[a 1 ,b 1 ]=[a 2 ,b 2 ] [c 1 ,d 1 ]=[c 2 ,d 2 ] a 1 b 2 = b 1 a 2 c 1 d 2 = d 1 c 2 <br />
<br />
(a 1 d 1 + b 1 c 1 )(b 2 d 2 )=a 1 d 1 b 2 d 2 + b 1 c 1 b 2 d 2<br />
= a 1 b 2 d 1 d 2 + b 1 b 2 c 1 d 2<br />
= b 1 a 2 d 1 d 2 + b 1 b 2 d 1 c 2<br />
=(b 1 d 1 )(a 2 d 2 + b 2 c 2 ).<br />
S F D ∼<br />
<br />
<br />
[a, b]+[c, d] =[ad + bc, bd]<br />
[a, b] · [c, d] =[ac, bd],
[0, 1] [1, 1] <br />
[0, 1] <br />
[a, b]+[0, 1] = [a1+b0,b1] = [a, b].<br />
[1, 1] [a, b] ∈ F D a ≠0<br />
[b, a] F D [a, b] · [b, a] =[1, 1] [b, a] <br />
[a, b] [−a, b] [a, b] <br />
F D <br />
F D <br />
F D <br />
<br />
[a, b][e, f]+[c, d][e, f] =[ae, bf]+[ce, df]<br />
=[aedf + bfce, bdf 2 ]<br />
=[aed + bce, bdf]<br />
=[ade + bce, bdf]<br />
=([a, b]+[c, d])[e, f]<br />
F D <br />
D<br />
D D <br />
F D F D <br />
D F D E <br />
D ψ : F D → E <br />
E ψ(a) =a a ∈ D a F D <br />
D F D <br />
φ : D → F D φ(a) =[a, 1] a b D<br />
<br />
φ(a + b) =[a + b, 1] = [a, 1]+[b, 1] = φ(a)+φ(b)<br />
φ(ab) =[ab, 1] = [a, 1][b, 1] = φ(a)φ(b);<br />
φ φ φ(a) =φ(b)<br />
[a, 1] = [b, 1] a = a1 =1b = b F D <br />
D <br />
φ(a)[φ(b)] −1 =[a, 1][b, 1] −1 =[a, 1] · [1,b]=[a, b].<br />
E D ψ : F D → E ψ([a, b]) = ab −1 <br />
ψ [a 1 ,b 1 ]=[a 2 ,b 2 ] a 1 b 2 = b 1 a 2 a 1 b −1<br />
1 =<br />
a 2 b −1<br />
2 ψ([a 1 ,b 1 ]) = ψ([a 2 ,b 2 ])<br />
[a, b] [c, d] F D <br />
ψ([a, b]+[c, d]) = ψ([ad + bc, bd])<br />
=(ad + bc)(bd) −1<br />
= ab −1 + cd −1<br />
= ψ([a, b]) + ψ([c, d])
ψ([a, b] · [c, d]) = ψ([ac, bd])<br />
=(ac)(bd) −1<br />
= ab −1 cd −1<br />
= ψ([a, b])ψ([c, d]).<br />
ψ <br />
ψ <br />
ψ([a, b]) = ab −1 =0 a =0b =0 [a, b] =[0,b] ψ <br />
[0,b] F D ψ <br />
Q Q[x] Q[x]<br />
p(x)/q(x) p(x) q(x) <br />
q(x) Q(x)<br />
<br />
F F <br />
Q<br />
F p F <br />
Z p <br />
<br />
<br />
F <br />
F [x] <br />
<br />
<br />
R a b R <br />
a b a | b c ∈ R b = ac <br />
R a b R <br />
u R a = ub<br />
D p ∈ D <br />
p = ab a b p <br />
p | ab p | a p | b<br />
<br />
R Q[x, y] x 2 y 2 xy <br />
R xy xy x 2 y 2 <br />
x 2 y 2 <br />
n>1 <br />
p 1 ···p k p i <br />
p i <br />
<br />
<br />
D D <br />
a ∈ D a ≠0 a a <br />
D
a = p 1 ···p r = q 1 ···q s p i q i r = s<br />
π ∈ S r p i q π(j) j =1,...,r<br />
<br />
<br />
<br />
Z[ √ 3 i] ={a + b √ 3 i} <br />
z = a + b √ 3 i ν : Z[ √ 3 i] → N ∪{0} ν(z) =|z| 2 = a 2 +3b 2 <br />
ν(z) ≥ 0 z =0 <br />
ν(zw) =ν(z)ν(w) ν(z) =1 z <br />
Z[ √ 3 i] 1 −1<br />
4 <br />
4=2· 2=(1− √ 3 i)(1 + √ 3 i).<br />
Z[ √ 3 i] 2 <br />
2=zw z,w Z[ √ 3 i] ν(z) =ν(w) =2 <br />
z Z[ √ 3 i] ν(z) =2 <br />
a 2 +3b 2 =2 2 <br />
1− √ 3 i 1+ √ 3 i 2 <br />
1 − √ 3 i 1+ √ 3 i 4 <br />
<br />
R <br />
a ∈ R 〈a〉 = {ra : r ∈ R} <br />
<br />
D a, b ∈ D <br />
a | b 〈b〉 ⊂〈a〉<br />
a b 〈b〉 = 〈a〉<br />
a D 〈a〉 = D<br />
a | b b = ax x ∈ D r <br />
D br =(ax)r = a(xr) 〈b〉 ⊂〈a〉 〈b〉 ⊂〈a〉 b ∈〈a〉<br />
b = ax x ∈ D a | b<br />
a b u a = ub <br />
b | a 〈a〉 ⊂〈b〉 〈b〉 ⊂〈a〉 〈a〉 = 〈b〉 <br />
〈a〉 = 〈b〉 a | b b | a a = bx b = ay x, y ∈ D<br />
a = bx = ayx D xy =1 x y <br />
a b <br />
a ∈ D a 1 a <br />
1 〈a〉 = 〈1〉 = D<br />
D 〈p〉 D 〈p〉 <br />
p <br />
〈p〉 a D p <br />
〈p〉 ⊂〈a〉 〈p〉 D = 〈a〉 〈p〉 = 〈a〉 a p<br />
a p <br />
p 〈a〉 D 〈p〉 ⊂〈a〉 ⊂D <br />
a | p p a a p <br />
D = 〈a〉 〈p〉 = 〈a〉 〈p〉
D p p <br />
p p | ab 〈ab〉 ⊂〈p〉 <br />
〈p〉 〈p〉 a ∈〈p〉 b ∈〈p〉<br />
p | a p | b<br />
D I 1 ,I 2 ,... I 1 ⊂ I 2 ⊂ ···<br />
N I n = I N n ≥ N<br />
I = ⋃ ∞<br />
i=1 I i D I I 1 ⊂ I<br />
0 ∈ I a, b ∈ I a ∈ I i b ∈ I j i j N <br />
i ≤ j a b I j a − b <br />
I j r ∈ D a ∈ I a ∈ I i i <br />
I i ra ∈ I i I I <br />
D<br />
D a ∈ D I<br />
a I N N ∈ N I N = I = 〈a〉 I n = I N <br />
n ≥ N<br />
<br />
<br />
<br />
<br />
D a D <br />
a <br />
a = a 1 b 1 a 1 b 1 〈a〉 ⊂〈a 1 〉 <br />
〈a〉 ≠ 〈a 1 〉 a a 1 b 1 <br />
a 1 = a 2 b 2 a 2 b 2 <br />
〈a 1 〉⊂〈a 2 〉 <br />
<br />
〈a〉 ⊂〈a 1 〉⊂〈a 2 〉⊂···.<br />
N 〈a n 〉 = 〈a N 〉 n ≥ N<br />
a N a <br />
<br />
a = c 1 p 1 p 1 c 1 <br />
〈a〉 ⊂〈c 1 〉 c 1 c 1 = c 2 p 2 <br />
p 2 c 2 <br />
<br />
〈a〉 ⊂〈c 1 〉⊂〈c 2 〉⊂···.<br />
<br />
a = p 1 p 2 ···p r<br />
p 1 ,...,p r <br />
<br />
a = p 1 p 2 ···p r = q 1 q 2 ···q s ,
p i q i <br />
r
z w Q(i) ={p + qi : p, q ∈ Q} Z[i]<br />
<br />
zw −1 =(a + bi) c − di<br />
c 2 + d 2<br />
ac + bd bc − ad<br />
=<br />
c 2 +<br />
+ d2 c 2 + d 2 i<br />
(<br />
= m 1 + n ) (<br />
1<br />
c 2 + d 2 + m 2 + n )<br />
2<br />
c 2 + d 2 i<br />
(<br />
n1<br />
=(m 1 + m 2 i)+<br />
c 2 + d 2 + n )<br />
2<br />
c 2 + d 2 i<br />
=(m 1 + m 2 i)+(s + ti)<br />
Q(i) <br />
m i <br />
|n i /(a 2 + b 2 )|≤1/2 <br />
9<br />
8 =1+1 8<br />
15<br />
8 =2− 1 8 .<br />
s t zw −1 =(m 1 + m 2 i)+(s + ti) <br />
s 2 + t 2 ≤ 1/4 + 1/4 = 1/2 w <br />
z = zw −1 w = w(m 1 + m 2 i)+w(s + ti) =qw + r,<br />
q = m 1 + m 2 i r = w(s + ti) z qw Z[i] r Z[i]<br />
r =0 ν(r)
Z[x] p(x) =5x 4 −3x 3 +x−4 <br />
1 q(x) =<br />
4x 2 − 6x +8 q(x) 2<br />
D f(x) g(x) <br />
D[x] f(x)g(x) <br />
f(x) = ∑ m<br />
i=0 a ix i g(x) = ∑ n<br />
i=0 b ix i p <br />
f(x)g(x) r p ∤ a r s <br />
p ∤ b s x r+s f(x)g(x) <br />
c r+s = a 0 b r+s + a 1 b r+s−1 + ···+ a r+s−1 b 1 + a r+s b 0 .<br />
p a 0 ,...,a r−1 b 0 ,...,b s−1 p c r+s <br />
a r b s p | c r+s p a r p b s <br />
D p(x) q(x) D[x] <br />
p(x)q(x) p(x) q(x)<br />
p(x) =cp 1 (x) q(x) =dq 1 (x) c d p(x)<br />
q(x) p 1 (x) q 1 (x) p(x)q(x) =<br />
cdp 1 (x)q 1 (x) p 1 (x)q 1 (x) p(x)q(x) cd<br />
D F p(x) ∈ D[x] <br />
p(x) =f(x)g(x) f(x) g(x) F [x] p(x) =f 1 (x)g 1 (x) f 1 (x)<br />
g 1 (x) D[x] f(x) = f 1 (x) g(x) = g 1 (x)<br />
a b D af(x),bg(x) D[x] <br />
a 1 ,b 2 ∈ D af(x) =a 1 f 1 (x) bg(x) =b 1 g 1 (x) f 1 (x) g 1 (x)<br />
D[x] abp(x) =(a 1 f 1 (x))(b 1 g 1 (x)) f 1 (x)<br />
g 1 (x) ab | a 1 b 1 <br />
c ∈ D p(x) =cf 1 (x)g 1 (x) f(x) = f 1 (x) <br />
g(x) = g 1 (x)<br />
<br />
D F p(x)<br />
D[x] F [x] D[x]<br />
D F p(x) <br />
D[x] p(x) =f(x)g(x) F [x] p(x) =f 1 (x)g 1 (x) f 1 (x) g 1 (x) <br />
D[x] f(x) = f 1 (x) g(x) = g 1 (x)<br />
D D[x] <br />
p(x) D[x] p(x) <br />
D p(x) <br />
D[x] F D <br />
p(x) =f 1 (x)f 2 (x) ···f n (x) p(x) f i (x) <br />
a i ∈ D a i f i (x) D[x] b 1 ,...,b n ∈ D a i f i (x) =b i g i (x)<br />
g i (x) D[x] g i (x) <br />
D[x] <br />
a 1 ···a n p(x) =b 1 ···b n g 1 (x) ···g n (x).
= b 1 ···b n g 1 (x) ···g n (x) a 1 ···a n b p(x) =<br />
ag 1 (x) ···g n (x) a ∈ D D a uc 1 ···c k u <br />
c i D<br />
<br />
p(x) =a 1 ···a m f 1 (x) ···f n (x) =b 1 ···b r g 1 (x) ···g s (x)<br />
p(x) D[x] <br />
f i g i F [x] a i b i <br />
F F [x] n = s g i (x) <br />
f i (x) g i (x) i =1,...,n c 1 ,...,c n d 1 ,...,d n <br />
D (c i /d i )f i (x) =g i (x) c i f i (x) =d i g i (x) f i (x) g i (x) <br />
c i d i D a 1 ···a m = ub 1 ···b r D u <br />
D D m = s <br />
b i a i b i i <br />
<br />
<br />
F F [x] <br />
Z[x] <br />
D D[x 1 ,...,x n ] <br />
<br />
<br />
<br />
Z[x]<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
17 <br />
n <br />
N =2 2n +1 N <br />
<br />
<br />
<br />
<br />
i √ −1
z = a + b √ 3 i Z[ √ 3 i] a 2 +3b 2 =1 z <br />
Z[ √ 3 i] 1 −1<br />
Z[i] Z[i]<br />
<br />
5<br />
1+3i<br />
6+8i<br />
2<br />
<br />
D <br />
F D <br />
F D <br />
F D <br />
F 1 <br />
D D <br />
D<br />
F F <br />
Q<br />
F <br />
F [x] F (x) <br />
p(x)/q(x) q(x) <br />
p(x 1 ,...,x n ) q(x 1 ,...,x n ) F [x 1 ,...,x n ] <br />
p(x 1 ,...,x n )/q(x 1 ,...,x n ) <br />
F [x 1 ,...,x n ] F [x 1 ,...,x n ] <br />
F (x 1 ,...,x n )
p Z p [x] Z p (x) Z p (x)<br />
p<br />
Z[i] <br />
Q(i) ={p + qi : p, q ∈ Q}.<br />
F E <br />
F E E F <br />
<br />
F F <br />
Q<br />
F p F <br />
Z p <br />
<br />
Z[ √ 2]={a + b √ 2:a, b ∈ Z}<br />
Z[ √ 2] <br />
Z[ √ 2]<br />
Z[ √ 2]<br />
Z[ √ 2i] ν(a+b √ 2 i) =<br />
a 2 +2b 2 <br />
D d ∈ D a b <br />
D d | a d | b d a b<br />
D a b D <br />
a b d d ′ <br />
a b d d ′ (a, b)<br />
a b<br />
D a b D <br />
s t D (a, b) =as + bt<br />
D D a ∼ b a b <br />
D ∼ D<br />
D ν u D <br />
ν(u) =ν(1)<br />
D ν a b <br />
D ν(a) =ν(b)<br />
Z[ √ 5 i] <br />
<br />
R <br />
a 1 ,...,a n R r ∈ R a 1 r 1 + ···+ a n r n<br />
r 1 ,...,r n R R <br />
R <br />
D I 1 ⊃ I 2 ⊃ I 3 ⊃ ···<br />
N I k = I N k ≥ N
D <br />
<br />
R <br />
R S 1 ∈ S ab ∈ S a, b ∈ S<br />
∼ R × S (a, s) ∼ (a ′ ,s ′ ) s ∗ ∈ S <br />
s ∗ (s ′ a − sa ′ )=0 ∼ R × S<br />
a/s (a, s) ∈ R × S S −1 R <br />
∼ <br />
S −1 R <br />
a<br />
s + b t<br />
a<br />
s<br />
=<br />
at + bs<br />
st<br />
b<br />
t = ab<br />
st ,<br />
S −1 R S −1 R<br />
S −1 R <br />
R S<br />
ψ : R → S −1 R ψ(a) =a/1 <br />
R 0/∈ S ψ <br />
P R S = R \ P <br />
R<br />
P R S = R \ P S −1 R
19<br />
<br />
<br />
<br />
<br />
<br />
<br />
N Z Q R <br />
<br />
<br />
<br />
<br />
<br />
<br />
X X × X P X <br />
X <br />
(a, a) ∈ P a ∈ X<br />
(a, b) ∈ P (b, a) ∈ P a = b<br />
(a, b) ∈ P (b, c) ∈ P (a, c) ∈ P <br />
a ≼ b (a, b) ∈ P <br />
a ≤ b a b A ⊂ B <br />
A B X ≼ <br />
<br />
a ≤ b <br />
a b Z<br />
X X <br />
X X P(X) X = {a, b, c}<br />
P(X) {a, b, c}<br />
∅ {a} {b} {c}<br />
{a, b} {a, c} {b, c} {a, b, c}.<br />
X ⊂ <br />
{a, b, c}
{a, b, c}<br />
{a, b} {a, c} {b, c}<br />
{a} {b} {c}<br />
∅<br />
P({a, b, c})<br />
G G <br />
<br />
<br />
N a ≼ b a | b a | a <br />
a ∈ N m | n n | m m = n <br />
m | n n | p m | p<br />
X = {1, 2, 3, 4, 6, 8, 12, 24} 24 <br />
X<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
24<br />
Y X u X Y a ≼ u<br />
a ∈ Y u Y u ≼ v <br />
v Y u Y l <br />
X Y l ≼ a a ∈ Y l Y <br />
k ≼ l k Y l <br />
Y <br />
Y = {2, 3, 4, 6} X Y<br />
12 24 12 1<br />
<br />
<br />
Y X Y <br />
Y Y Y
u 1 u 2 Y <br />
u 1 ≼ u u Y u 1 ≼ u 2 u 2 ≼ u 1 <br />
u 1 = u 2 <br />
<br />
<br />
L <br />
L <br />
a, b ∈ L a b a ∨ b <br />
a, b ∈ L a b a ∧ b<br />
X X P(X) <br />
A B P(X) A B A ∪ B A ∪ B <br />
A B A ⊂ A ∪ B B ⊂ A ∪ B C <br />
A B C A ∪ B A ∪ B A B<br />
A B A ∩ B<br />
G X G <br />
X ⊂ G <br />
H K G H K H ∩ K<br />
H ∪ K G <br />
H K H ∪ K<br />
<br />
(A ∪ B) ′ A ′ ∩ B ′ <br />
<br />
<br />
≼ ≽ ∨ ∧ <br />
<br />
<br />
L ∨ ∧ <br />
a, b, c ∈ L<br />
a ∨ b = b ∨ a a ∧ b = b ∧ a<br />
a ∨ (b ∨ c) =(a ∨ b) ∨ c a ∧ (b ∧ c) =(a ∧ b) ∧ c<br />
a ∨ a = a a ∧ a = a<br />
a ∨ (a ∧ b) =a a ∧ (a ∨ b) =a<br />
<br />
a ∨ b {a, b} b ∨ a <br />
{b, a} {a, b} = {b, a}<br />
a ∨ (b ∨ c) (a ∨ b) ∨ c {a, b, c}<br />
d = a ∨ b c ≼ d ∨ c =(a ∨ b) ∨ c <br />
a ≼ a ∨ b = d ≼ d ∨ c =(a ∨ b) ∨ c.<br />
b ≼ (a ∨ b) ∨ c (a ∨ b) ∨ c <br />
{a, b, c} (a ∨ b) ∨ c {a, b, c}<br />
u {a, b, c} a ≼ u b ≼ u d = a ∨ b ≼ u
c ≼ u (a ∨ b) ∨ c = d ∨ c ≼ u (a ∨ b) ∨ c <br />
{a, b, c} a ∨ (b ∨ c) <br />
{a, b, c} a ∨ (b ∨ c) =(a ∨ b) ∨ c<br />
a a {a} a ∨ a = a<br />
d = a ∧ b a ≼ a ∨ d d = a ∧ b ≼ a a ∨ d ≼ a<br />
a ∨ (a ∧ b) =a<br />
L ∨ ∧ <br />
<br />
<br />
L ∨ ∧ <br />
<br />
L a ≼ b a ∨ b = b L ≼ <br />
a, b ∈ L a b a ∨ b <br />
a ∧ b <br />
L ≼ a∨a = a a ≼ a ≼ <br />
≼ a ≼ b b ≼ a a ∨ b = b b ∨ a = a <br />
b = a ∨ b = b ∨ a = a ≼ <br />
a ≼ b b ≼ c a ∨ b = b b ∨ c = c <br />
a ∨ c = a ∨ (b ∨ c) =(a ∨ b) ∨ c = b ∨ c = c,<br />
a ≼ c<br />
L a ∨ b a ∧ b <br />
a b a =(a ∨ b) ∧ a = a ∧ (a ∨ b) <br />
a ≼ a ∨ b b ≼ a ∨ b a ∨ b a b u<br />
a b a ≼ u b ≼ u a ∨ b ≼ u <br />
(a ∨ b) ∨ u = a ∨ (b ∨ u) =a ∨ u = u.<br />
a ∧ b a b <br />
<br />
<br />
P(X) X <br />
<br />
P(X) X ∅ A<br />
P(X) A ∩ X = A A ∪∅= A <br />
I X a ≼ I a ∈ X <br />
O X O ≼ a a ∈ X<br />
A P(X) A <br />
A ′ = X \ A = {x : x ∈ X x /∈ A}.<br />
A ∪ A ′ = X A ∩ A ′ = ∅ <br />
L I O <br />
a ∈ L a ′ a ∨ a ′ = I a ∧ a ′ = O<br />
L ∨ ∧ <br />
<br />
a ∧ (b ∨ c) =(a ∧ b) ∨ (a ∧ c);
P(X) <br />
A ∩ (B ∪ C) =(A ∩ B) ∪ (A ∩ C)<br />
A, B, C ∈P(X) L <br />
<br />
a ∧ (b ∨ c) =(a ∧ b) ∨ (a ∧ c)<br />
a, b, c ∈ L<br />
L <br />
a, b, c ∈ L<br />
a ∨ (b ∧ c) =(a ∨ b) ∧ (a ∨ c)<br />
<br />
L <br />
a ∨ (b ∧ c) =[a ∨ (a ∧ c)] ∨ (b ∧ c)<br />
= a ∨ [(a ∧ c) ∨ (b ∧ c)]<br />
= a ∨ [(c ∧ a) ∨ (c ∧ b)]<br />
= a ∨ [c ∧ (a ∨ b)]<br />
= a ∨ [(a ∨ b) ∧ c]<br />
=[(a ∨ b) ∧ a] ∨ [(a ∨ b) ∧ c]<br />
=(a ∨ b) ∧ (a ∨ c).<br />
<br />
B I <br />
O B X P(X) <br />
<br />
<br />
∨ ∧ <br />
<br />
B <br />
∨ ∧ B <br />
a ∨ b = b ∨ a a ∧ b = b ∧ a a, b ∈ B<br />
a ∨ (b ∨ c) =(a ∨ b) ∨ c a ∧ (b ∧ c) =(a ∧ b) ∧ c a, b, c ∈ B<br />
a ∧ (b ∨ c) =(a ∧ b) ∨ (a ∧ c) a ∨ (b ∧ c) =(a ∨ b) ∧ (a ∨ c) a, b, c ∈ B<br />
I O a ∨ O = a a ∧ I = a a ∈ B<br />
a ∈ B a ′ ∈ B a ∨ a ′ = I a ∧ a ′ = O<br />
B <br />
<br />
a = a ∨ O<br />
= a ∨ (a ∧ a ′ )<br />
=(a ∨ a) ∧ (a ∨ a ′ )
=(a ∨ a) ∧ I<br />
= a ∨ a.<br />
I ∨ b =(b ∨ b ′ ) ∨ b =(b ′ ∨ b) ∨ b = b ′ ∨ (b ∨ b) =b ′ ∨ b = I.<br />
<br />
a ∨ (a ∧ b) =(a ∧ I) ∨ (a ∧ b)<br />
= a ∧ (I ∨ b)<br />
= a ∧ I<br />
= a.<br />
B <br />
B <br />
B <br />
a ∈ B O ∨ a = a O ≼ a O B <br />
I B a ∨ b = b a ∧ b = a<br />
a ∨ I = a a ∈ B <br />
a ∨ I =(a ∧ I) ∨ I = I ∨ (I ∧ a) =I<br />
a ≼ I a B B B <br />
<br />
B I O <br />
B a ∨ b a ∧ b <br />
{a, b} B <br />
<br />
<br />
<br />
B <br />
a ∨ I = I a ∧ O = O a ∈ B<br />
a ∨ b = a ∨ c a ∧ b = a ∧ c a, b, c ∈ B b = c<br />
a ∨ b = I a ∧ b = O b = a ′ <br />
(a ′ ) ′ = a a ∈ B<br />
I ′ = O O ′ = I<br />
(a ∨ b) ′ = a ′ ∧ b ′ (a ∧ b) ′ = a ′ ∨ b ′ <br />
<br />
a ∨ b = a ∨ c a ∧ b = a ∧ c <br />
b = b ∨ (b ∧ a)<br />
= b ∨ (a ∧ b)<br />
= b ∨ (a ∧ c)<br />
=(b ∨ a) ∧ (b ∨ c)
=(a ∨ b) ∧ (b ∨ c)<br />
=(a ∨ c) ∧ (b ∨ c)<br />
=(c ∨ a) ∧ (c ∨ b)<br />
= c ∨ (a ∧ b)<br />
= c ∨ (a ∧ c)<br />
= c ∨ (c ∧ a)<br />
= c.<br />
<br />
<br />
<br />
B C φ : B → C <br />
<br />
φ(a ∨ b) =φ(a) ∨ φ(b)<br />
φ(a ∧ b) =φ(a) ∧ φ(b)<br />
a b B<br />
<br />
X <br />
B <br />
a ∈ B B a ≠ O a ∧ b = a b ∈ B a <br />
B b ∈ B a O ≼ b ≼ a<br />
B b B <br />
a B a ≼ b<br />
b a = b b 1 O b<br />
b 1 ≼ b b b 1 <br />
b 2 O b 1 b 2 ≼ b 1 <br />
b 2 a = b 2 <br />
O ≼···≼b 3 ≼ b 2 ≼ b 1 ≼ b.<br />
B k b k <br />
a = b k <br />
a b B a ≠ b <br />
a ∧ b = O<br />
a ∧ b a b a ∧ b ≼ a <br />
a ∧ b = a a ∧ b = O a ∧ b = a a ≼ b a = O <br />
a b a ∧ b = O<br />
B a, b ∈ B <br />
<br />
a ≼ b<br />
a ∧ b ′ = O
a ′ ∨ b = I<br />
⇒ a ≼ b a ∨ b = b <br />
a ∧ b ′ = a ∧ (a ∨ b) ′<br />
= a ∧ (a ′ ∧ b ′ )<br />
=(a ∧ a ′ ) ∧ b ′<br />
= O ∧ b ′<br />
= O.<br />
⇒ a ∧ b ′ = O a ′ ∨ b =(a ∧ b ′ ) ′ = O ′ = I<br />
⇒ a ′ ∨ b = I <br />
a ≼ b<br />
a = a ∧ (a ′ ∨ b)<br />
=(a ∧ a ′ ) ∨ (a ∧ b)<br />
= O ∨ (a ∧ b)<br />
= a ∧ b.<br />
B b c B b ⋠ c<br />
a ∈ B a ≼ b a ⋠ c<br />
b ∧ c ′ ≠ O a a ≼ b ∧ c ′ <br />
a ≼ b a ⋠ c<br />
b ∈ B a 1 ,...,a n B a i ≼ b <br />
b = a 1 ∨···∨a n a, a 1 ,...,a n B a ≼ b a i ≼ b <br />
b = a ∨ a 1 ∨···∨a n a = a i i =1,...,n<br />
b 1 = a 1 ∨···∨a n a i ≼ b i b 1 ≼ b <br />
b ≼ b 1 b ⋠ b 1 <br />
a a ≼ b a ⋠ b 1 a a ≼ b <br />
a = a i a i a ≼ b 1 b ≼ b 1 <br />
b = a 1 ∨···∨a n a b<br />
a = a ∧ b = a ∧ (a 1 ∨···∨a n )=(a ∧ a 1 ) ∨···∨(a ∧ a n ).<br />
O a a ∧ a i a i <br />
a = a i i<br />
B X <br />
B P(X)<br />
B P(X) X B<br />
a ∈ B a a = a 1 ∨···∨a n a 1 ,...,a n ∈ X<br />
φ : B →P(X) <br />
φ(a) =φ(a 1 ∨···∨a n )={a 1 ,...,a n }.<br />
φ <br />
a = a 1 ∨···∨a n b = b 1 ∨···∨b m B a i <br />
b i φ(a) =φ(b) {a 1 ,...,a n } = {b 1 ,...,b m } a = b φ<br />
<br />
a b φ <br />
φ(a ∨ b) =φ(a 1 ∨···∨a n ∨ b 1 ∨···∨b m )
φ(a ∧ b) =φ(a) ∩ φ(b)<br />
= {a 1 ,...,a n ,b 1 ,...,b m }<br />
= {a 1 ,...,a n }∪{b 1 ,...,b m }<br />
= φ(a 1 ∨···∨a n ) ∪ φ(b 1 ∧···∨b m )<br />
= φ(a) ∪ φ(b).<br />
<br />
2 n <br />
n<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
a <br />
<br />
a<br />
b <br />
A B <br />
a b a ∧ b <br />
a b <br />
A B <br />
a b a ∨ b<br />
A a b B<br />
a ∧ b<br />
a<br />
A<br />
B<br />
b<br />
a ∨ b
a∧b b∧a <br />
<br />
∨ ∧ <br />
a ∧ (b ∨ c) =(a ∧ b) ∨ (a ∧ c) <br />
a a ′ a <br />
a I <br />
O a ∧ a ′ = O a ∨ a ′ = I <br />
b<br />
a<br />
b<br />
a<br />
c<br />
a<br />
c<br />
a ∧ (b ∨ c) =(a ∧ b) ∨ (a ∧ c)<br />
a a ′ a<br />
a ∧ a ′ = O a ∨ a ′ = I<br />
a ′<br />
<br />
(a ∨ b) ∧ (a ∨ b ′ ) ∧ (a ∨ b) <br />
<br />
<br />
<br />
<br />
<br />
(a ∨ b) ∧ (a ∨ b ′ ) ∧ (a ∨ b) =(a ∨ b) ∧ (a ∨ b) ∧ (a ∨ b ′ )<br />
=(a ∨ b) ∧ (a ∨ b ′ )<br />
= a ∨ (b ∧ b ′ )<br />
= a ∨ O<br />
= a,<br />
a<br />
<br />
a<br />
a<br />
a<br />
b<br />
b ′<br />
b<br />
(a ∨ b) ∧ (a ∨ b ′ ) ∧ (a ∨ b)
X = {a, b, c, d} <br />
⊂<br />
30 <br />
<br />
Z 12 <br />
B 36 B <br />
a ≼ b a | b B X B <br />
P(X)<br />
Z a ≼ b a | b<br />
<br />
(a ∨ b ∨ a ′ ) ∧ a<br />
(a ∨ b) ′ ∧ (a ∨ b)<br />
a ∨ (a ∧ b)<br />
(c ∨ a ∨ b) ∧ c ′ ∧ (a ∨ b) ′<br />
a b c
a<br />
a b c<br />
a c ′ a<br />
a ′ b<br />
b<br />
c ′<br />
X n P(X) =2 n <br />
2 n n ∈ N<br />
<br />
<br />
<br />
a ′ a b ′<br />
b<br />
a a b<br />
a ′<br />
b a ′ b<br />
a b c<br />
a ′ b ′ c<br />
a b ′ c ′<br />
a ≼ b <br />
a | b<br />
L ∨ ∧ <br />
L<br />
a ≼ b a ∨ b = b a b<br />
a ∧ b<br />
G X G <br />
H K G H K <br />
H ∪ K<br />
R X R X <br />
⊂ I J X I ∩ J<br />
I J I + J R
B <br />
a ∨ I = I a ∧ O = O a ∈ B<br />
a ∨ b = I a ∧ b = O b = a ′ <br />
(a ′ ) ′ = a a ∈ B<br />
I ′ = O O ′ = I<br />
(a ∨ b) ′ = a ′ ∧ b ′ (a ∧ b) ′ = a ′ ∨ b ′ <br />
<br />
<br />
B + · B <br />
a + b =(a ∧ b ′ ) ∨ (a ′ ∧ b)<br />
a · b = a ∧ b.<br />
B a 2 = a a ∈ B<br />
X a b X a ≼ b b ≼ a X <br />
<br />
a | b N<br />
N Z Q R ≤<br />
X Y φ : X → Y a ≼ b <br />
φ(a) ≼ φ(b) L M ψ : L → M <br />
ψ(a ∨ b) =ψ(a) ∨ ψ(b) ψ(a ∧ b) =ψ(a) ∧ ψ(b) <br />
<br />
<br />
B a = b (a ∧ b ′ ) ∨ (a ′ ∧ b) =O <br />
a, b ∈ B<br />
B a = O (a ∧ b ′ ) ∨ (a ′ ∧ b) =b <br />
b ∈ B<br />
L M L × M (a, b) ≼ (c, d) a ≼ c<br />
b ≼ d L × M <br />
<br />
<br />
n f : {O, I} n →{0,I}<br />
<br />
x 1 ,...,x n O I<br />
∨ ∧ ′ <br />
<br />
x y x ′ x ∨ y x∧ y<br />
0 0 1 0 0<br />
0 1 1 1 0<br />
1 0 0 1 0<br />
1 1 0 1 1
20<br />
<br />
<br />
<br />
<br />
x y z <br />
<br />
n <br />
<br />
<br />
<br />
<br />
<br />
<br />
V F α · v αv<br />
α ∈ F v ∈ V <br />
α(βv) =(αβ)v<br />
(α + β)v = αv + βv<br />
α(u + v) =αu + αv<br />
1v = v<br />
α, β ∈ F u, v ∈ V <br />
V F <br />
<br />
<br />
<br />
<br />
<br />
n R n R<br />
u =(u 1 ,...,u n ) v =(v 1 ,...,v n ) R n α R <br />
<br />
<br />
u + v =(u 1 ,...,u n )+(v 1 ,...,v n )=(u 1 + v 1 ,...,u n + v n )<br />
αu = α(u 1 ,...,u n )=(αu 1 ,...,αu n ).
F F [x] F F [x]<br />
α ∈ F <br />
p(x) ∈ F [x] αp(x)<br />
[a, b] <br />
R f(x) g(x) [a, b] (f + g)(x) <br />
f(x) +g(x) (αf)(x) =αf(x) α ∈ R <br />
f(x) = x g(x) =x 2 (2f +5g)(x) =2 x +5x 2 <br />
V = Q( √ 2) = {a + b √ 2:a, b ∈ Q} V <br />
Q u = a + b √ 2 v = c + d √ 2 u + v =(a + c)+(b + d) √ 2 V <br />
α ∈ Q αv V <br />
V <br />
V F <br />
<br />
0v = v ∈ V <br />
α = α ∈ F <br />
αv = α =0 v = <br />
(−1)v = −v v ∈ V <br />
−(αv) =(−α)v = α(−v) α ∈ F v ∈ V <br />
<br />
<br />
0v =(0+0)v =0v +0v;<br />
+0v =0v +0v V =0v<br />
α =0<br />
α ≠0 αv = 1/α v = <br />
<br />
v +(−1)v =1v +(−1)v =(1− 1)v =0v = ,<br />
−v =(−1)v <br />
<br />
<br />
<br />
V F W V W <br />
V u, v ∈ W <br />
α ∈ F u + v αv W <br />
W R 3 W = {(x 1 , 2x 1 + x 2 ,x 1 − x 2 ):<br />
x 1 ,x 2 ∈ R} W R 3 <br />
α(x 1 , 2x 1 + x 2 ,x 1 − x 2 )=(αx 1 ,α(2x 1 + x 2 ),α(x 1 − x 2 ))<br />
=(αx 1 , 2(αx 1 )+αx 2 ,αx 1 − αx 2 ),<br />
W W <br />
u =(x 1 , 2x 1 + x 2 ,x 1 − x 2 ) v =(y 1 , 2y 1 + y 2 ,y 1 − y 2 ) W <br />
u + v =(x 1 + y 1 , 2(x 1 + y 1 )+(x 2 + y 2 ), (x 1 + y 1 ) − (x 2 + y 2 )).
W F [x] <br />
p(x) q(x) p(x)+q(x) αp(x) ∈ W <br />
α ∈ F p(x) ∈ W <br />
V F v 1 ,v 2 ,...,v n <br />
V α 1 ,α 2 ,...,α n F w V <br />
w =<br />
n∑<br />
α i v i = α 1 v 1 + α 2 v 2 + ···+ α n v n<br />
i=1<br />
v 1 ,v 2 ,...,v n <br />
v 1 ,v 2 ,...,v n <br />
v 1 ,v 2 ,...,v n W v 1 ,v 2 ,...,v n W <br />
v 1 ,v 2 ,...,v n <br />
S = {v 1 ,v 2 ,...,v n } V <br />
S V <br />
u v S <br />
v i <br />
<br />
u = α 1 v 1 + α 2 v 2 + ···+ α n v n<br />
v = β 1 v 1 + β 2 v 2 + ···+ β n v n .<br />
u + v =(α 1 + β 1 )v 1 +(α 2 + β 2 )v 2 + ···+(α n + β n )v n<br />
v i α ∈ F <br />
S<br />
αu =(αα 1 )v 1 +(αα 2 )v 2 + ···+(αα n )v n<br />
<br />
<br />
S = {v 1 ,v 2 ,...,v n } V <br />
α 1 ,α 2 ...α n ∈ F α i <br />
<br />
α 1 v 1 + α 2 v 2 + ···+ α n v n = ,<br />
S S <br />
S <br />
<br />
{α 1 ,α 2 ...α n }<br />
α 1 v 1 + α 2 v 2 + ···+ α n v n = <br />
α 1 = α 2 = ···= α n =0<br />
{v 1 ,v 2 ,...,v n } <br />
<br />
v = α 1 v 1 + α 2 v 2 + ···+ α n v n = β 1 v 1 + β 2 v 2 + ···+ β n v n .<br />
α 1 = β 1 ,α 2 = β 2 ,...,α n = β n
v = α 1 v 1 + α 2 v 2 + ···+ α n v n = β 1 v 1 + β 2 v 2 + ···+ β n v n ,<br />
(α 1 − β 1 )v 1 +(α 2 − β 2 )v 2 + ···+(α n − β n )v n = .<br />
v 1 ,...,v n α i − β i =0 i =1,...,n<br />
<br />
<br />
{v 1 ,v 2 ,...,v n } V <br />
v i <br />
{v 1 ,v 2 ,...,v n } <br />
α 1 ,...,α n <br />
α 1 v 1 + α 2 v 2 + ···+ α n v n = ,<br />
α i α k ≠0 <br />
v k = − α 1<br />
v 1 −···− α k−1<br />
α k<br />
<br />
α k<br />
v k−1 − α k+1<br />
α k<br />
v k+1 −···− α n<br />
α k<br />
v n .<br />
v k = β 1 v 1 + ···+ β k−1 v k−1 + β k+1 v k+1 + ···+ β n v n .<br />
<br />
β 1 v 1 + ···+ β k−1 v k−1 − v k + β k+1 v k+1 + ···+ β n v n = .<br />
<br />
<br />
<br />
V n m>n<br />
m V <br />
{e 1 ,e 2 ,...,e n } V V {e 1 ,e 2 ,...,e n }<br />
V <br />
e 1 =(1, 0, 0) e 2 =(0, 1, 0) e 3 =(0, 0, 1) <br />
R 3 R 3 (x 1 ,x 2 ,x 3 ) R 3 <br />
x 1 e 1 + x 2 e 2 + x 3 e 3 e 1 ,e 2 ,e 3 <br />
e 1 ,e 2 ,e 3<br />
R 3 {(3, 2, 1), (3, 2, 0), (1, 1, 1)} R 3 <br />
Q( √ 2)={a+b √ 2:a, b ∈ Q} {1, √ 2 } {1+ √ 2, 1− √ 2 }<br />
Q( √ 2)<br />
<br />
<br />
R 3 <br />
Q( √ 2) <br />
<br />
{e 1 ,e 2 ,...,e m } {f 1 ,f 2 ,...,f n } <br />
V m = n
{e 1 ,e 2 ,...,e m } <br />
n ≤ m {f 1 ,f 2 ,...,f n } <br />
m ≤ n m = n<br />
{e 1 ,e 2 ,...,e n } V <br />
V n V = n <br />
<br />
V n<br />
S = {v 1 ,...,v n } V S <br />
V <br />
S = {v 1 ,...,v n } V S V <br />
S = {v 1 ,...,v k } V k
{(x 1 ,x 2 ,x 3 ):3x 1 +4x 3 =0, 2x 1 − x 2 + x 3 =0}<br />
{(x 1 ,x 2 ,x 3 ):x 1 − 2x 2 +2x 3 =2}<br />
{(x 1 ,x 2 ,x 3 ):3x 1 − 2x 2 2 =0}<br />
<br />
(x, y, z) ∈ R 3 <br />
Ax + By + Cz =0<br />
Dx + Ey + Cz =0<br />
R 3 <br />
W [0, 1] f(0) = 0 <br />
W C[0, 1]<br />
V F −(αv) =(−α)v = α(−v) α ∈ F<br />
v ∈ V <br />
V n <br />
S = {v 1 ,...,v n } V S <br />
V <br />
S = {v 1 ,...,v n } V S V <br />
S = {v 1 ,...,v k } V kn<br />
V W F <br />
m n T : V → W <br />
T (u + v) =T (u)+T (v)<br />
T (αv) =αT (v)<br />
α ∈ F u, v ∈ V T V W <br />
T (T )={v ∈ V : T (v) =} V <br />
T T <br />
T R(V )={w ∈ W : T (v) =w v ∈<br />
V } W <br />
T : V → W (T )={}<br />
{v 1 ,...,v k } T <br />
{v 1 ,...,v k ,v k+1 ,...,v m } V {T (v k+1 ),...,T(v m )} <br />
T T m − k
V = W T : V → W <br />
<br />
V W n F <br />
T : V → W {v 1 ,...,v n } V <br />
{T (v 1 ),...,T(v n )} W <br />
F n F n <br />
U V W U <br />
V U + V u + v u ∈ U<br />
v ∈ V <br />
U + V U ∩ V W <br />
U + V = W U ∩ V = W <br />
W = U ⊕ V w ∈ W <br />
w = u + v u ∈ U v ∈ V <br />
U k W n <br />
V n − k W = U ⊕ V <br />
V <br />
U V W <br />
(U + V )= U + V − (U ∩ V ).<br />
<br />
V W F <br />
V W (V,W)<br />
F <br />
(S + T )(v) =S(v)+T (v)<br />
(αS)(v) =αS(v),<br />
S, T ∈ (V,W) α ∈ F v ∈ V <br />
V F V V ∗ = (V,F)<br />
V v 1 ,...,v n <br />
V v = α 1 v 1 + ···+ α n v n V <br />
φ i : V → F φ i (v) =α i φ i V ∗ <br />
v 1 ,...,v n <br />
<br />
{(3, 1), (2, −2)} R 2 (R 2 ) ∗ <br />
V n F V ∗∗ <br />
V ∗ v ∈ V λ v V ∗∗ <br />
v ↦→ λ v V V ∗∗
21<br />
<br />
F <br />
<br />
Q R <br />
<br />
F p(x) ∈ F [x] <br />
E F p(x) <br />
E[x] <br />
p(x) =x 4 − 5x 2 +6<br />
Q[x] p(x) (x 2 − 2)(x 2 − 3) <br />
Q[x] p(x) <br />
<br />
p(x) =(x − √ 2)(x + √ 2)(x − √ 3)(x + √ 3).<br />
p(x) <br />
Q( √ 2) = {a + b √ 2:a, b ∈ Q}.<br />
<br />
F <br />
<br />
<br />
E F F E F <br />
F ⊂ E<br />
<br />
F = Q( √ 2)={a + b √ 2:a, b ∈ Q}<br />
E = Q( √ 2+ √ 3) Q √ 2+ √ 3 <br />
E F E <br />
F √ 2 E √ 2+ √ 3 E<br />
1/( √ 2+ √ 3) = √ 3 − √ 2 E √ 2+ √ 3 <br />
√<br />
3 −<br />
√<br />
2 <br />
√<br />
2 <br />
√<br />
3 E<br />
p(x) =x 2 + x +1 ∈ Z 2 [x] <br />
p(x) Z 2 <br />
Z 2 α p(α) =0 〈p(x)〉
p(x) Z 2 [x]/〈p(x)〉 <br />
Z 2 [x]/〈p(x)〉 <br />
f(x) +〈p(x)〉 <br />
f(x) =(x 2 + x +1)q(x)+r(x),<br />
r(x) x 2 + x +1 <br />
f(x)+〈x 2 + x +1〉 = r(x)+〈x 2 + x +1〉.<br />
r(x) 0 1 x 1+x E = Z 2 [x]/〈x 2 +x+1〉<br />
Z 2 α p(x)<br />
Z 2 (α) <br />
0+0α =0<br />
1+0α =1<br />
0+1α = α<br />
1+1α =1+α.<br />
α 2 + α +1=0 (1 + α) 2 <br />
(1 + α)(1 + α) =1+α + α +(α) 2 = α.<br />
<br />
E<br />
+ 0 1 α 1+α<br />
0 0 1 α 1+α<br />
1 1 0 1+α α<br />
α α 1+α 0 1<br />
1+α 1+α α 1 0<br />
Z 2 (α)<br />
· 0 1 α 1+α<br />
0 0 0 0 0<br />
1 0 1 α 1+α<br />
α 0 α 1+α 1<br />
1+α 0 1+α 1 α<br />
Z 2 (α)<br />
<br />
<br />
F p(x) F [x] <br />
E F α ∈ E p(α) =0
p(x) <br />
E F α p(α) =0 〈p(x)〉<br />
p(x) F [x] F [x]/〈p(x)〉 <br />
E = F [x]/〈p(x)〉 <br />
E F <br />
ψ : F → F [x]/〈p(x)〉 ψ(a) =a + 〈p(x)〉 a ∈ F <br />
ψ <br />
ψ(a)+ψ(b) =(a + 〈p(x)〉)+(b + 〈p(x)〉) =(a + b)+〈p(x)〉 = ψ(a + b)<br />
<br />
ψ(a)ψ(b) =(a + 〈p(x)〉)(b + 〈p(x)〉) =ab + 〈p(x)〉 = ψ(ab).<br />
ψ <br />
a + 〈p(x)〉 = ψ(a) =ψ(b) =b + 〈p(x)〉.<br />
a−b p(x) 〈p(x)〉 p(x) <br />
a − b =0 a = b ψ <br />
ψ F {a + 〈p(x)〉 : a ∈ F } E <br />
E F <br />
p(x) α ∈ E α = x + 〈p(x)〉 α <br />
E p(x) =a 0 + a 1 x + ···+ a n x n <br />
p(α) =a 0 + a 1 (x + 〈p(x)〉)+···+ a n (x + 〈p(x)〉) n<br />
= a 0 +(a 1 x + 〈p(x)〉)+···+(a n x n + 〈p(x)〉)<br />
= a 0 + a 1 x + ···+ a n x n + 〈p(x)〉<br />
=0+〈p(x)〉.<br />
α ∈ E = F [x]/〈p(x)〉 α p(x)<br />
p(x) =x 5 +x 4 +1 ∈ Z 2 [x] p(x) x 2 +x+1<br />
x 3 + x +1 E Z 2 p(x) E E<br />
Z 2 [x]/〈x 2 + x +1〉 Z 2 [x]/〈x 3 + x +1〉 <br />
Z 2 [x]/〈x 3 + x +1〉 2 3 =8<br />
<br />
α E F F f(α) =0 <br />
f(x) ∈ F [x] E F <br />
F E F F <br />
E F E F α 1 ,...,α n E<br />
F α 1 ,...,α n F (α 1 ,...,α n ) E = F (α)<br />
α ∈ E E F <br />
√ 2 i Q <br />
x 2 −2 x 2 +1 π e <br />
Q R <br />
Q Q <br />
<br />
π + e <br />
[0, 1]
Q <br />
C Q<br />
√ 2+ √ 3 Q α = √ 2+ √ 3 <br />
α 2 =2+ √ 3 α 2 − 2= √ 3 (α 2 − 2) 2 =3 α 4 − 4α 2 +1=0 <br />
α x 4 − 4x 2 +1∈ Q[x]<br />
E F E <br />
F <br />
<br />
E F α ∈ E α <br />
F F (α) F (x) F [x]<br />
φ α : F [x] → E α α <br />
F φ α (p(x)) = p(α) ≠ 0 <br />
p(x) ∈ F [x] φ α = {0} φ α <br />
E F [x] F [x] <br />
F (x) E <br />
<br />
E F α ∈ E α <br />
F p(x) ∈ F [x] <br />
p(α) =0f(x) F [x] f(α) =0 p(x) <br />
f(x)<br />
φ α : F [x] → E φ α <br />
p(x) ∈ F [x] p(x) ≥ 1 <br />
F [x] α 〈p(x)〉<br />
F [x] α f(α) =0 f(x) <br />
f(x) ∈〈p(x)〉 p(x) f(x) p(x) <br />
α α <br />
βp(x) β ∈ F <br />
p(x) =r(x)s(x) p(x) <br />
p(α) =0 r(α)s(α) =0 r(α) =0 s(α) =0 <br />
p p(x) <br />
E F α ∈ E F <br />
p(x) α F <br />
p(x) α F <br />
f(x) =x 2 − 2 g(x) =x 4 − 4x 2 +1 <br />
√ 2 √ 2+ √ 3 <br />
E F α ∈ E F <br />
F (α) ∼ = F [x]/〈p(x)〉 p(x) α F <br />
φ α : F [x] → E <br />
〈p(x)〉 p(x) α <br />
φ α E F (α) F α<br />
E = F (α) F α ∈ E <br />
F α F n β ∈ E <br />
<br />
β = b 0 + b 1 α + ···+ b n−1 α n−1<br />
b i ∈ F
φ α (F [x]) ∼ = F (α) E = F (α) <br />
φ α (f(x)) = f(α) f(α) α F <br />
p(x) =x n + a n−1 x n−1 + ···+ a 0<br />
α p(α) =0 <br />
<br />
α n+1 = αα n<br />
α n = −a n−1 α n−1 −···−a 0 .<br />
= −a n−1 α n − a n−2 α n−1 −···−a 0 α<br />
= −a n−1 (−a n−1 α n−1 −···−a 0 ) − a n−2 α n−1 −···−a 0 α.<br />
α m m ≥ n <br />
α n β ∈ F (α) <br />
<br />
b i c i F <br />
β = b 0 + b 1 α + ···+ b n−1 α n−1 .<br />
β = b 0 + b 1 α + ···+ b n−1 α n−1 = c 0 + c 1 α + ···+ c n−1 α n−1<br />
g(x) =(b 0 − c 0 )+(b 1 − c 1 )x + ···+(b n−1 − c n−1 )x n−1<br />
F [x] g(α) =0 g(x) p(x) <br />
α g(x) <br />
b 0 − c 0 = b 1 − c 1 = ···= b n−1 − c n−1 =0,<br />
b i = c i i =0, 1,...,n− 1 <br />
x 2 +1 R 〈x 2 +1〉 R[x] <br />
E = R[x]/〈x 2 +1〉 R x 2 +1 α = x+〈x 2 +1〉<br />
E E <br />
R(α) ={a + bα : a, b ∈ R} α 2 = −1 E <br />
α 2 +1=(x + 〈x 2 +1〉) 2 +(1+〈x 2 +1〉)<br />
=(x 2 +1)+〈x 2 +1〉<br />
=0.<br />
R(α) C a + bα <br />
a + bi<br />
E F E F <br />
<br />
E F <br />
E <br />
E F <br />
E F F
E = F (α) F <br />
{1,α,α 2 ,...,α n−1 }<br />
E F F <br />
n E n F <br />
E F <br />
[E : F ]=n.<br />
E F <br />
<br />
α ∈ E [E : F ]=n <br />
1,α,...,α n<br />
a i ∈ F <br />
<br />
p(α) =0<br />
a n α n + a n−1 α n−1 + ···+ a 1 α + a 0 =0.<br />
p(x) =a n x n + ···+ a 0 ∈ F [x]<br />
F <br />
<br />
R Q Q<br />
<br />
<br />
<br />
E F K E K<br />
F <br />
[K : F ]=[K : E][E : F ].<br />
{α 1 ,...,α n } E F {β 1 ,...,β m } <br />
K E {α i β j } K F <br />
K u ∈ K u = ∑ m<br />
j=1 b jβ j b j = ∑ n<br />
i=1 a ijα i <br />
b j ∈ E a ij ∈ F <br />
(<br />
m∑ n∑<br />
)<br />
u = a ij α i β j = ∑ a ij (α i β j ).<br />
j=1 i=1<br />
i,j<br />
mn α i β j K F <br />
{α i β j } <br />
v 1 ,v 2 ,...,v n V <br />
<br />
c 1 v 1 + c 2 v 2 + ···+ c n v n =0<br />
<br />
<br />
c 1 = c 2 = ···= c n =0.<br />
u = ∑ c ij (α i β j )=0<br />
i,j
c ij ∈ F c ij u <br />
m∑<br />
j=1<br />
( n∑<br />
i=1<br />
c ij α i<br />
)<br />
β j =0,<br />
∑ i c ijα i ∈ E β j E <br />
<br />
n∑<br />
c ij α i =0<br />
i=1<br />
j α j F c ij =0 i<br />
j <br />
<br />
F i i =1,...,k F i+1 F i <br />
F k F 1 <br />
[F k : F 1 ]=[F k : F k−1 ] ···[F 2 : F 1 ].<br />
E F α ∈ E F <br />
p(x) β ∈ F (α) q(x) q(x) <br />
p(x)<br />
p(x) =[F (α) :F ] q(x) =[F (β) :F ] F ⊂<br />
F (β) ⊂ F (α)<br />
[F (α) :F ]=[F (α) :F (β)][F (β) :F ].<br />
Q √ 3+ √ 5 <br />
√ 3+ √ 5 x 4 − 16x 2 +4 <br />
[Q( √ 3+ √ 5):Q] =4.<br />
{1, √ 3 } Q( √ √ √<br />
3) Q 3+ 5 <br />
Q( √ 3) √ 5 Q( √ 3) {1, √ 5 } <br />
Q( √ 3, √ 5) = (Q( √ 3 ))( √ 5) Q( √ 3) {1, √ 3, √ 5, √ 3 √ 5= √ 15 } <br />
Q( √ 3, √ 5) = Q( √ 3+ √ 5) Q <br />
F (α 1 ,...,α n ) F n>1<br />
Q( 3√ 5, √ 5 i) √ 5 <br />
5 3√ 5 5 √ 5 i /∈ Q( 3√ 5)<br />
[Q( 3√ 5, √ 5 i) :Q( 3√ 5)]=2.<br />
{1, √ 5i } Q( 3√ 5, √ 5 i) Q( 3√ 5) <br />
{1, 3√ 5, ( 3√ 5) 2 } Q( 3√ 5) Q Q( 3√ 5, √ 5 i) Q <br />
{1, √ 5 i, 3√ 5, ( 3√ 5) 2 , ( 6√ 5) 5 i, ( 6√ 5) 7 i =5 6√ 5 i 6√ 5 i}.<br />
6√ 5 i x 6 +5 <br />
Q p =5 <br />
Q ⊂ Q( 6√ 5 i) ⊂ Q( 3√ 5, √ 5 i).<br />
Q( 6√ 5 i) =Q( 3√ 5, √ 5 i) <br />
6
E F <br />
<br />
<br />
E F <br />
α 1 ,...,α n ∈ E E =<br />
F (α 1 ,...,α n )<br />
<br />
E = F (α 1 ,...,α n ) ⊃ F (α 1 ,...,α n−1 ) ⊃···⊃F (α 1 ) ⊃ F,<br />
F (α 1 ,...,α i ) F (α 1 ,...,α i−1 )<br />
⇒ E F E <br />
F α 1 ,...,α n<br />
E E = F (α 1 ,...,α n ) α i F <br />
⇒ E = F (α 1 ,...,α n ) α i F <br />
E = F (α 1 ,...,α n ) ⊃ F (α 1 ,...,α n−1 ) ⊃···⊃F (α 1 ) ⊃ F,<br />
F (α 1 ,...,α i ) F (α 1 ,...,α i−1 )<br />
⇒ <br />
E = F (α 1 ,...,α n ) ⊃ F (α 1 ,...,α n−1 ) ⊃···⊃F (α 1 ) ⊃ F,<br />
F (α 1 ,...,α i ) F (α 1 ,...,α i−1 ) <br />
F (α 1 ,...,α i )=F (α 1 ,...,α i−1 )(α i )<br />
α i F (α 1 ,...,α i−1 ) <br />
[F (α 1 ,...,α i ):F (α 1 ,...,α i−1 )]<br />
i [E : F ] <br />
<br />
F E <br />
p(x) E <br />
E F <br />
F <br />
E <br />
α, β ∈ E F F (α, β) F <br />
F (α, β) F α ± β αβ α/β β ≠0 <br />
F E F <br />
<br />
Q <br />
E F F<br />
E E F F <br />
F [x] F <br />
F <br />
F [x] F [x]
F p(x) ∈ F [x] <br />
p(x) F α x − α p(x) <br />
p(x) =(x − α)q 1 (x) q 1 (x) = p(x) − 1 q 1 (x) <br />
<br />
p(x) =(x − α)(x − β)q 2 (x),<br />
q 2 (x) = p(x) − 2 p(x)<br />
<br />
p(x) F [x] <br />
ax − b p(b/a) =0 F <br />
<br />
F E<br />
E F F ⊂ E α ∈ E <br />
α x − α α ∈ F F = E<br />
F <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
F p(x) F [x] <br />
F p(x) <br />
E F p(x) <br />
F p(x) <br />
p(x)<br />
F p(x) =a 0 + a 1 x + ···+ a n x n F [x]<br />
E F p(x) α 1 ,...,α n <br />
E E = F (α 1 ,...,α n ) <br />
p(x) =(x − α 1 )(x − α 2 ) ···(x − α n ).<br />
p(x) ∈ F [x] E E[x]<br />
p(x) =x 4 +2x 2 − 8 Q[x] p(x) <br />
x 2 − 2 x 2 +4 Q( √ 2,i) p(x)<br />
p(x) =x 3 − 3 Q[x] p(x) Q( 3√ 3)<br />
p(x) <br />
Q( 3√ 3)<br />
− 3√ 3 ± ( 6√ 3) 5 i<br />
,<br />
2
p(x) ∈ F [x] <br />
E p(x)<br />
p(x) p(x) =1 <br />
p(x) E = F <br />
k 1 ≤ k
E[x]/〈p(x)〉<br />
σ<br />
E(α)<br />
ψ<br />
φ<br />
F [x]/〈q(x)〉<br />
τ<br />
F (β)<br />
E<br />
φ<br />
F<br />
<br />
φ : E → F p(x) <br />
E[x] q(x) F [x] <br />
K p(x) L q(x) φ <br />
ψ : K → L<br />
p(x) <br />
p(x) E q(x) F p(x) =1 <br />
K = E L = F <br />
n K <br />
p(x) p(x) K α <br />
E ⊂ E(α) ⊂ K β q(x) L F ⊂ F (β) ⊂ L<br />
φ : E(α) → F (β) φ(α) =β φ<br />
φ E<br />
K<br />
ψ<br />
L<br />
E(α)<br />
φ<br />
F (β)<br />
E<br />
φ<br />
F<br />
p(x) =(x − α)f(x) q(x) =(x − β)g(x) f(x) <br />
g(x) p(x) q(x) K
f(x) E(α) L g(x) F (β) <br />
ψ : K → L ψ φ <br />
E(α) ψ : K → L ψ φ E<br />
p(x) F [x] K<br />
p(x) <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
30 ◦ 90 ◦<br />
20 ◦ <br />
60 ◦ <br />
<br />
α |α| <br />
<br />
F <br />
<br />
α β α + β α − β αβ <br />
α/β β ≠0 α β <br />
α>β α + β α − β <br />
αβ β>1 <br />
△ABC △ADE α/1 = x/β x <br />
αβ β
D<br />
β<br />
B<br />
1<br />
α<br />
A<br />
x<br />
C<br />
E<br />
<br />
α √ α <br />
△ABD △BCD △ABC <br />
1/x = x/α x 2 = α<br />
B<br />
x<br />
A<br />
<br />
D<br />
α<br />
C<br />
<br />
P =(p, q) <br />
p q <br />
<br />
F R<br />
F ax + by + c =0 a<br />
b c F <br />
F <br />
F x 2 + y 2 + dx + ey + f =0 d e f F <br />
(x 1 ,y 1 ) (x 2 ,y 2 ) F <br />
x 1 = x 2 x − x 1 =0 <br />
ax + by + c =0 x 1 ≠ x 2 <br />
<br />
( )<br />
y2 − y 1<br />
y − y 1 =<br />
(x − x 1 ),<br />
x 2 − x 1<br />
<br />
(x 1 ,y 1 ) <br />
r <br />
(x − x 1 ) 2 +(y − y 1 ) 2 − r 2 =0.
F <br />
R <br />
R <br />
F <br />
F<br />
F F <br />
R<br />
R <br />
F F <br />
R <br />
ax + by + c =0 F F <br />
<br />
x 2 + y 2 + d 1 x + e 1 y + f 1 =0<br />
x 2 + y 2 + d 2 x + e 2 y + f 2 =0<br />
d i e i f i F i =1, 2 <br />
<br />
<br />
x 2 + y 2 + d 1 x + e 1 x + f 1 =0<br />
(d 1 − d 2 )x + b(e 2 − e 1 )y +(f 2 − f 1 )=0.<br />
<br />
<br />
<br />
<br />
<br />
ax + by + c =0<br />
x 2 + y 2 + dx + ey + f =0.<br />
y Ax 2 +Bx+C =0<br />
A B C F x <br />
x = −B ± √ B 2 − 4AC<br />
2A<br />
F ( √ α ) α = B 2 − 4AC > 0 <br />
F <br />
F F ( √ α ) α F <br />
α <br />
<br />
Q = F 0 ⊂ F 1 ⊂···⊂F k<br />
F i = F i−1 ( √ α i ) α i ∈ F i α ∈ F k <br />
k>0 [Q(α) :Q] =2 k
F i α i <br />
<br />
[F k : Q] =[F k : F k−1 ][F k−1 : F k−2 ] ···[F 1 : Q] =2 k .<br />
Q<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
1 <br />
1 2 <br />
3√ 2 <br />
3√<br />
2 x 3 − 2 Q<br />
<br />
[Q( 3√ 2):Q] =3<br />
3 2<br />
1 <br />
π √ π <br />
π √ π <br />
<br />
<br />
<br />
20 ◦ 60 ◦ <br />
<br />
3θ = (2θ + θ)<br />
= 2θ θ − 2θ θ<br />
=(2 2 θ − 1) θ − 2 2 θ θ<br />
=(2 2 θ − 1) θ − 2(1 − 2 θ) θ<br />
=4 3 θ − 3 θ.<br />
θ α = θ θ =20 ◦ <br />
3θ = 60 ◦ = 1/2 <br />
4α 3 − 3α = 1 2 .<br />
α 8x 3 − 6x − 1 Z[x] <br />
Q[x] [Q(α) :Q] =3 α
x 2 + y 2 = z 2 <br />
x n + y n = z n n ≥ 3 <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
p(x, y) Z[x, y] <br />
<br />
<br />
Q <br />
Q<br />
<br />
√<br />
1/3 + √ 7<br />
√ 3+ 3√ 5<br />
√ 3+ √ 2 i<br />
θ + i θ θ =2π/n n ∈ N<br />
√ 3 √ 2 − i
Q( √ 3, √ 6) Q<br />
Q( 3√ 2, 3√ 3) Q<br />
Q( √ 2,i) Q<br />
Q( √ 3, √ 5, √ 7) Q<br />
Q( √ 2, 3√ 2) Q<br />
Q( √ 8) Q( √ 2)<br />
Q(i, √ 2+i, √ 3+i) Q<br />
Q( √ 2+ √ 5) Q( √ 5)<br />
Q( √ 2, √ 6+ √ 10 ) Q( √ 3+ √ 5)<br />
<br />
<br />
x 4 − 10x 2 +21 Q<br />
x 4 +1 Q<br />
x 3 +2x +2 Z 3<br />
x 3 − 3 Q<br />
<br />
Q( 4√ 3,i) Q<br />
Q( 4√ 3,i) Q [Q( 4√ 3,i):Q] =8<br />
F Q( 4√ 3,i) [F : Q] =2<br />
F Q( 4√ 3,i) [F : Q] =4<br />
Z 2 [x]/〈x 3 + x +1〉 <br />
<br />
9 <br />
20 <br />
1 ◦ Q <br />
<br />
Q( √ 3, 4√ 3, 8√ 3,...) Q <br />
π Q(π 3 )<br />
p(x) n F [x] <br />
E p(x) [E : F ] ≤ n!<br />
Q( √ 2) ∼ = Q( √ 3)<br />
Q( 4√ 3) Q( 4√ 3 i) <br />
K E E F <br />
K F <br />
Z[x]/〈x 3 − 2〉 <br />
F p p(x) =x p − a <br />
F F
E F p(x) F [x]<br />
E<br />
p(x) F [x] F <br />
<br />
α β β ≠0 α/β<br />
R Q <br />
Q <br />
E F σ E<br />
F α ∈ E σ <br />
α E<br />
Q( √ 3, √ 7)=Q( √ 3+ √ 7) Q( √ a, √ b )=<br />
Q( √ a + √ b ) (a, b) =1<br />
E F [E : F ]=2 E <br />
F f(x) ∈ F [x]<br />
p(x) Z 6 [x] <br />
R p(x) R<br />
E F α ∈ E [F (α) :F (α 3 )]<br />
α, β Q αβ α+β <br />
E F α ∈ E F <br />
F (α) F F <br />
α p(x) ∈ F [x] p = n<br />
[F (α) :F ]=n
22<br />
<br />
<br />
Z p p <br />
p n p n <br />
<br />
<br />
<br />
<br />
F p p <br />
α F pα =0 F <br />
0 p F <br />
n nα =0 α F <br />
F p p n <br />
<br />
F F p p <br />
p <br />
<br />
F p F p n <br />
n ∈ N<br />
φ : Z → F φ(n) =n · 1 <br />
F p φ pZ φ <br />
F Z p K F <br />
K K <br />
[F : K] =n F F K <br />
α 1 ,...,α n ∈ F α F <br />
α = a 1 α 1 + ···+ a n α n ,<br />
a i K p K p n <br />
α i F p n <br />
p D <br />
p <br />
a pn + b pn =(a + b) pn<br />
n
n <br />
n =1 <br />
p∑<br />
( p<br />
(a + b) p = a<br />
k)<br />
k b p−k .<br />
0
p n <br />
F p n p n <br />
x pn − x Z p <br />
f(x) =x pn − x F f(x) <br />
f(x) p n F f ′ (x) =p n x pn−1 − 1=−1 <br />
f(x) f(x) F <br />
f(x) α β f(x) α + β αβ f(x) <br />
α pn +β pn =(α+β) pn α pn β pn =(αβ) pn <br />
f(x) f(x) α f(x)<br />
−α f(x) <br />
f(−α) =(−α) pn − (−α) =−α pn + α = −(α pn − α) =0,<br />
p p =2 <br />
f(−α) =(−α) 2n − (−α) =α + α =0.<br />
α ≠0 (α −1 ) pn =(α pn ) −1 = α −1 f(x) F <br />
f(x) F <br />
E p n E F <br />
E f(x) f(x) α <br />
E E p n − 1<br />
α pn−1 =1 α pn − α =0 E p n E <br />
f(x) <br />
<br />
p n p n <br />
(p n )<br />
(p n ) p m m <br />
n m | n m>0 (p n ) <br />
(p m )<br />
F E = (p n ) F K <br />
p m K Z p m | n [E : K] =[E : F ][F :<br />
K]<br />
m | n m>0 p m − 1 p n − 1<br />
x pm−1 − 1 x pn−1 − 1 x pm − x x pn − x <br />
x pm − x x pn − x (p n ) <br />
x pm − x (p m )<br />
(p 24 )
(p 24 )<br />
(p 8 )<br />
(p 12 )<br />
(p 4 )<br />
(p 6 )<br />
(p 2 )<br />
(p 3 )<br />
(p)<br />
(p 24 )<br />
F F <br />
F ∗ <br />
<br />
G F ∗ <br />
F G <br />
G F ∗ n <br />
<br />
G ∼ = Z e p 1 ×···×Z e<br />
1 p k ,<br />
k<br />
n = p e 1<br />
1 ···pe k<br />
k<br />
p 1 ,...,p k m <br />
p e 1<br />
1 ,...,pe k<br />
k<br />
G m <br />
α G x r − 1 r m α x m − 1 x m − 1<br />
m F n ≤ m m ≤|G| <br />
m = n G n <br />
<br />
E F F <br />
α E ∗ E <br />
E = F (α)<br />
(2 4 ) Z 2 /〈1+x + x 4 〉 <br />
(2 4 ) <br />
{a 0 + a 1 α + a 2 α 2 + a 3 α 3 : a i ∈ Z 2 1+α + α 4 =0}.<br />
1+α + α 4 =0 (2 4 ) <br />
(2 4 ) Z 15<br />
α<br />
α 1 = α α 6 = α 2 + α 3 α 11 = α + α 2 + α 3<br />
α 2 = α 2 α 7 =1+α + α 3 α 12 =1+α + α 2 + α 3<br />
α 3 = α 3 α 8 =1+α 2 α 13 =1+α 2 + α 3<br />
α 4 =1+α α 9 = α + α 3 α 14 =1+α 3<br />
α 5 = α + α 2 α 10 =1+α + α 2 α 15 =1.
(n, k) <br />
E : Z k 2 → Zn 2 D : Zn 2 →<br />
Z k 2 D <br />
H ∈ M k×n (Z 2 )<br />
φ : Z k 2 → Zn 2 <br />
(n, k) φ (a 1 ,a 2 ,...,a n )<br />
n (a n ,a 1 ,a 2 ,...,a n−1 ) <br />
<br />
(6, 3) <br />
⎛ ⎞<br />
1 0 0<br />
0 1 0<br />
0 0 1<br />
G 1 =<br />
1 0 0<br />
⎜ ⎟<br />
⎝0 1 0⎠<br />
0 0 1<br />
<br />
⎛ ⎞<br />
1 0 0<br />
1 1 0<br />
1 1 1<br />
G 2 =<br />
.<br />
1 1 1<br />
⎜ ⎟<br />
⎝0 1 1⎠<br />
0 0 1<br />
(000) ↦→ (000000) (100) ↦→ (100100)<br />
(001) ↦→ (001001) (101) ↦→ (101101)<br />
(010) ↦→ (010010) (110) ↦→ (110110)<br />
(011) ↦→ (011011) (111) ↦→ (111111).<br />
<br />
<br />
(000) ↦→ (000000) (100) ↦→ (111100)<br />
(001) ↦→ (001111) (101) ↦→ (110011)<br />
(010) ↦→ (011110) (110) ↦→ (100010)<br />
(011) ↦→ (010001) (111) ↦→ (101101).<br />
(101101) (011011) <br />
<br />
<br />
Z 2 n <br />
Z 2 [x] n (a 0 ,a 1 ,...,a n−1 )<br />
<br />
f(x) =a 0 + a 1 x + ···+ a n−1 x n−1 ,<br />
f(x) n − 1 <br />
5 (10011) <br />
1+0x +0x 2 +1x 3 +1x 4 =1+x 3 + x 4 .
f(x) ∈ Z 2 [x] f(x)
f(t) t <br />
R n <br />
C Z n 2 R n =<br />
Z[x]/〈x n − 1〉<br />
C f(t) C tf(t) <br />
C t k f(t) C k ∈ N C <br />
f(t),tf(t),t 2 f(t),...,t n−1 f(t) <br />
p(t) p(t)f(t) C C <br />
C Z 2 [x]/〈x n +1〉 f(t) =a 0 + a 1 t + ···+<br />
a n−1 t n−1 C tf(t) C (a 1 ,...,a n−1 ,a 0 ) <br />
C<br />
R n <br />
Z n 2 R n <br />
φ : Z 2 [x] → R n φ[f(x)] = f(t) <br />
φ x n − 1 C R n <br />
φ(I) I Z 2 [x] 〈x n − 1〉 <br />
I Z 2 [x] Z 2 I = 〈g(x)〉<br />
Z 2 [x] 〈x n − 1〉 I <br />
g(x) x n − 1 C R n <br />
C = 〈g(t)〉 = {f(t)g(t) :f(t) ∈ R n g(x) | (x n − 1) Z 2 [x]}.<br />
C <br />
C<br />
x 7 − 1 <br />
x 7 − 1=(1+x)(1 + x + x 3 )(1 + x 2 + x 3 ).<br />
g(t) =(1+t + t 3 ) C R 7 (7, 4) <br />
g(t) <br />
t t 2 t 3 C <br />
⎛ ⎞<br />
1 0 0 0<br />
1 1 0 0<br />
0 1 1 0<br />
G =<br />
1 0 1 1<br />
.<br />
⎜<br />
0 1 0 1<br />
⎟<br />
⎝0 0 1 0⎠<br />
0 0 0 1<br />
(n, k) C <br />
t k x n − 1=g(x)h(x) Z 2 [x] g(x) =g 0 + g 1 x +<br />
···+ g n−k x n−k h(x) =h 0 + h 1 x + ···+ h k x k n × k <br />
⎛<br />
⎞<br />
g 0 0 ··· 0<br />
g 1 g 0 ··· 0<br />
<br />
G =<br />
g n−k g n−k−1 ··· g 0<br />
0 g n−k ··· g 1<br />
⎜<br />
<br />
⎝ ⎟ ⎠<br />
0 0 ··· g n−k
C g(t)<br />
C (n − k) × n <br />
⎛<br />
⎞<br />
0 ··· 0 0 h k ··· h 0<br />
0 ··· 0 h<br />
H =<br />
k ··· h 0 0<br />
⎜<br />
⎟<br />
⎝··· ··· ··· ··· ··· ··· ··· ⎠ .<br />
h k ··· h 0 0 0 ··· 0<br />
<br />
<br />
C = 〈g(t)〉 R n x n − 1=<br />
g(x)h(x) G H C <br />
HG =0<br />
<br />
x 7 − 1=g(x)h(x) =(1+x + x 3 )(1 + x + x 2 + x 4 ).<br />
<br />
⎛<br />
0 0 1 0 1 1<br />
⎞<br />
1<br />
H = ⎝0 1 0 1 1 1 0⎠ .<br />
1 0 1 1 1 0 0<br />
<br />
α 1 ,...,α n F <br />
n × n <br />
⎛<br />
⎞<br />
1 1 ··· 1<br />
α 1 α 2 ··· α n<br />
α 1 2 α2 2 ··· αn<br />
2 ⎜<br />
<br />
⎝ ⎟ ⎠<br />
α1 n−1 α2 n−1 ··· αn<br />
n−1<br />
<br />
<br />
<br />
α 1 ,...,α n F n ≥ 2 <br />
⎛<br />
⎞<br />
1 1 ··· 1<br />
α 1 α 2 ··· α n<br />
<br />
α 1 2 α2 2 ··· αn<br />
2 = ∏<br />
(α i − α j ).<br />
⎜<br />
<br />
⎝ ⎟ <br />
1≤j
p(x) <br />
n − 1 p(x) α 1 ,...,α n−1 <br />
<br />
<br />
<br />
<br />
<br />
p(x) =(x − α 1 )(x − α 2 ) ···(x − α n−1 )β,<br />
⎛<br />
⎞<br />
1 1 ··· 1<br />
α 1 α 2 ··· α n−1<br />
β =(−1) n+n <br />
α 1 2 α2 2 ··· αn−1<br />
2 .<br />
⎜<br />
<br />
⎝ ⎟ ⎠<br />
α1 n−2 α2 n−2 ··· αn−1<br />
n−2<br />
β =(−1) n+n<br />
∏<br />
1≤j
(ω i 0<br />
) r+1 x 0 +(ω i 1<br />
) r+1 x 1 + ···+(ω i s−1<br />
) r+1 x n−1 =0<br />
<br />
(ω i 0<br />
) r+s−1 x 0 +(ω i 1<br />
) r+s−1 x 1 + ···+(ω i s−1<br />
) r+s−1 x n−1 =0.<br />
<br />
⎛<br />
(ω i 0<br />
) r (ω i 1<br />
) r ··· (ω i s−1<br />
) r ⎞<br />
(ω i 0<br />
) r+1 (ω i 1<br />
) r+1 ··· (ω i s−1<br />
) r+1<br />
⎜<br />
⎝<br />
⎟<br />
<br />
⎠<br />
(ω i 0<br />
) r+s−1 (ω i 1<br />
) r+s−1 ··· (ω i s−1<br />
) r+s−1<br />
<br />
a i0 = a i1 = ···= a is−1 =0<br />
<br />
<br />
<br />
<br />
231 24 <br />
231 + 24 = 255 = 2 8 − 1 (255, 231) <br />
1 16 <br />
<br />
<br />
d =2r +1 <br />
r ≥ 0 ω n Z 2 m i (x) <br />
Z 2 ω i <br />
g(x) =[m 1 (x),m 2 (x),...,m 2r (x)],<br />
〈g(t)〉 R n n d <br />
C d<br />
C = 〈g(t)〉 R n <br />
<br />
<br />
C d<br />
f(t) C f(ω i )=0 1 ≤ i
m i (x) m i (x) ω i g(x) | f(x)<br />
g(x) f(x) <br />
⇒ f(t) =a 0 + a 1 t + ···+ a n−1 vt n−1 R n n<br />
Z n 2 =(a 0a 1 ···a n−1 ) <br />
⎛<br />
a 0 + a 1 ω + ···+ a n−1 ω n−1 ⎞ ⎛ ⎞<br />
f(ω)<br />
a 0 + a 1 ω 2 + ···+ a n−1 (ω 2 ) n−1<br />
H = ⎜<br />
⎟<br />
⎝<br />
<br />
⎠ = f(ω 2 )<br />
⎜ ⎟<br />
⎝ ⎠ =0<br />
a 0 + a 1 ω 2r + ···+ a n−1 (ω 2r ) n−1 f(ω 2r )<br />
f(t) C H C<br />
⇒ f(t) =a 0 +a 1 t+···+a n−1 t n−1 C <br />
f(ω i )=0 i =1,...,2r g(t) =[m 1 (t),...,m 2r (t)]<br />
C = 〈g(t)〉<br />
x 15 − 1 ∈ Z 2 [x] <br />
x 15 − 1=(x +1)(x 2 + x +1)(x 4 + x +1)(x 4 + x 3 +1)(x 4 + x 3 + x 2 + x +1),<br />
ω 1+x + x 4 <br />
(2 4 ) <br />
{a 0 + a 1 ω + a 2 ω 2 + a 3 ω 3 : a i ∈ Z 2 1+ω + ω 4 =0}.<br />
ω 15 ω <br />
m 1 (x) =1+x + x 4 ω 2 ω 4 m 1 (x) <br />
ω 3 m 2 (x) =1+x + x 2 + x 3 + x 4 <br />
g(x) =m 1 (x)m 2 (x) =1+x 4 + x 6 + x 7 + x 8<br />
ω ω 2 ω 3 ω 4 m 1 (x) m 2 (x) x 15 − 1 <br />
(15, 7) x 15 − 1=g(x)h(x) h(x) =1+x 4 + x 6 + x 7 <br />
<br />
⎛<br />
⎜<br />
⎝<br />
0 0 0 0 0 0 0 1 1 0 1 0 0 0 1<br />
0 0 0 0 0 0 1 1 0 1 0 0 0 1 0<br />
0 0 0 0 0 1 1 0 1 0 0 0 1 0 0<br />
0 0 0 0 1 1 0 1 0 0 0 1 0 0 0<br />
0 0 0 1 1 0 1 0 0 0 1 0 0 0 0<br />
0 0 1 1 0 1 0 0 0 1 0 0 0 0 0<br />
0 1 1 0 1 0 0 0 1 0 0 0 0 0 0<br />
1 1 0 1 0 0 0 1 0 0 0 0 0 0 0<br />
<br />
<br />
<br />
⎞<br />
.<br />
⎟<br />
⎠
[(3 6 ):(3 3 )]<br />
[(128) : (16)]<br />
[(625) : (25)]<br />
[(p 12 ):(p 2 )]<br />
[(p m ):(p n )] n | m<br />
(p 30 )<br />
α x 3 + x 2 +1 Z 2 8 <br />
x 3 + x 2 +1 Z 2 (α)<br />
27<br />
Q ∗ <br />
Z 2 [x]<br />
x 5 − 1<br />
x 6 + x 5 + x 4 + x 3 + x 2 + x +1<br />
x 9 − 1<br />
x 4 + x 3 + x 2 + x +1<br />
Z 2 [x]/〈x 3 + x +1〉 ∼ = Z 2 [x]/〈x 3 + x 2 +1〉<br />
n n =6, 7, 8, 10<br />
〈t +1〉 R n Z n 2 <br />
<br />
<br />
7<br />
15<br />
<br />
p Z p (x) <br />
p<br />
D p (a − b) pn = a pn − b pn <br />
a, b ∈ D<br />
<br />
E F K E F <br />
E = F <br />
F ⊂ E ⊂ K K F K <br />
E<br />
E F F q α ∈ E <br />
F n F (α) q n <br />
F E <br />
F α ∈ E E = F (α)<br />
n n Z p [x]<br />
Φ:(p n ) → (p n ) Φ:α ↦→ α p <br />
n<br />
(p n ) a p <br />
a ∈ (p n )<br />
E F (p n )|E| = p r |F | = p s <br />
E ∩ F <br />
p (p − 1)! ≡−1( p)
g(t) C R n <br />
g(x) 1<br />
<br />
<br />
<br />
C (n, k) n − k <br />
R n Z n 2 <br />
C R n g(t) 〈f(t)〉 R n <br />
〈g(t)〉 ⊂〈f(t)〉 f(x) g(x) Z 2 [x]<br />
C = 〈g(t)〉 R n x n − 1=g(x)h(x) <br />
g(x) =g 0 + g 1 x + ···+ g n−k x n−k h(x) =h 0 + h 1 x + ···+ h k x k G <br />
n × k <br />
⎛<br />
⎞<br />
g 0 0 ··· 0<br />
g 1 g 0 ··· 0<br />
<br />
G =<br />
g n−k g n−k−1 ··· g 0<br />
0 g n−k ··· g 1<br />
⎜<br />
<br />
⎝ ⎟ ⎠<br />
0 0 ··· g n−k<br />
H (n − k) × n <br />
⎛<br />
⎞<br />
0 ··· 0 0 h k ··· h 0<br />
0 ··· 0 h<br />
H =<br />
k ··· h 0 0<br />
⎜<br />
⎟<br />
⎝··· ··· ··· ··· ··· ··· ··· ⎠ .<br />
h k ··· h 0 0 0 ··· 0<br />
G C<br />
H C<br />
HG =0<br />
<br />
<br />
C R n <br />
c(t) =c 0 + c 1 t + ···+ c n−1 t n−1 <br />
w(t) =w 0 + w 1 t + ···w n−1 t n−1 R n <br />
a 1 ,...,a k w(t) =c(t)+e(t) e(t) =t a 1<br />
+ t a 2<br />
+ ···+ t a k<br />
<br />
a i c(t) <br />
w(t) a i w(t) w(ω i )=s i i =1,...,2r <br />
ω n Z 2 w(t) s 1 ,...,s 2r <br />
w(t) s i =0 i<br />
<br />
s i = w(ω i )=e(ω i )=ω ia 1<br />
+ ω ia 2<br />
+ ···+ ω ia k<br />
i =1,...,2r <br />
s(x) =(x + ω a 1<br />
)(x + ω a 2<br />
) ···(x + ω a k<br />
).
(15, 7) <br />
a 1 a 2 <br />
s(x) =(x + ω a 1<br />
)(x + ω a 2<br />
) <br />
(<br />
s(x) =x 2 + s 1 x + s 2 1 + s )<br />
3<br />
.<br />
s 1<br />
w(t) =1+t 2 + t 4 + t 5 + t 7 + t 12 + t 13
23<br />
<br />
<br />
<br />
<br />
<br />
x 5 − 1=0 x 6 − x 3 − 6=0 <br />
<br />
ax 5 + bx 4 + cx 3 + dx 2 + ex + f =0.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
F <br />
<br />
σ τ F στ σ −1 <br />
F <br />
<br />
E F <br />
E F σ : E → E <br />
σ(α) =α α ∈ F <br />
E F <br />
E σ τ <br />
E σ(α) =α τ(α) =α α ∈ F στ(α) =σ(α) =α <br />
σ −1 (α) =α E E<br />
F E
E F E<br />
(E) E F <br />
E F <br />
G(E/F )={σ ∈ (E) :σ(α) =α α ∈ F }.<br />
f(x) F [x] E f(x) F <br />
f(x) G(E/F )<br />
σ : a + bi ↦→ a − bi <br />
<br />
σ(a) =σ(a +0i) =a − 0i = a,<br />
G(C/R)<br />
Q ⊂ Q( √ 5)⊂ Q( √ 3, √ 5) a, b ∈ Q( √ 5)<br />
σ(a + b √ 3)=a − b √ 3<br />
Q( √ 3, √ 5) Q( √ 5) <br />
τ(a + b √ 5)=a − b √ 5<br />
Q( √ 3, √ 5) Q( √ 3) μ = στ <br />
√ 3 √ 5 {,σ,τ,μ} Q( √ 3, √ 5)<br />
Q Z 2 × Z 2 <br />
σ τ μ<br />
σ τ μ<br />
σ σ μ τ<br />
τ τ μ σ<br />
μ μ τ σ <br />
Q( √ 3, √ 5) Q {1, √ 3, √ 5, √ 15 }<br />
|G(Q( √ 3, √ 5)/Q)| =[Q( √ 3, √ 5):Q)] = 4<br />
E F f(x) F [x] <br />
G(E/F ) f(x) E<br />
<br />
<br />
f(x) =a 0 + a 1 x + a 2 x 2 + ···+ a n x n<br />
α ∈ E f(x) σ ∈ G(E/F )<br />
σ(α) f(x)<br />
0=σ(0)<br />
= σ(f(α))<br />
= σ(a 0 + a 1 α + a 2 α 2 + ···+ a n α n )<br />
= a 0 + a 1 σ(α)+a 2 [σ(α)] 2 + ···+ a n [σ(α)] n ;<br />
E F α, β ∈ E <br />
F Q( √ √ √<br />
2) <br />
2 − 2 Q <br />
x 2 − 2
α β F σ :<br />
F (α) → F (β) σ F <br />
f(x) F [x] E <br />
f(x) F f(x) <br />
|G(E/F )| =[E : F ].<br />
f(x) f(x)<br />
0 1 E = F <br />
k 0 ≤ k
f(x) =x 4 + x 3 + x 2 + x +1<br />
Q f(x) <br />
(x − 1)f(x) =x 5 − 1 <br />
f(x) ω i i =1,...,4 <br />
ω = (2π/5) + i (2π/5).<br />
f(x) Q(ω) σ i Q(ω) <br />
σ i (ω) =ω i i =1,...,4 <br />
G(Q(ω)/Q) <br />
[Q(ω) :Q] =|G(Q(ω)/Q)| =4,<br />
σ i G(Q(ω)/Q) G(Q(ω)/Q) ∼ = Z 4 ω <br />
<br />
<br />
f(x) <br />
F [x] <br />
E <br />
f(x) F [x] f(x) E <br />
r∏<br />
f(x) =(x − α 1 ) n 1<br />
(x − α 2 ) n2 ···(x − α r ) nr = (x − α i ) n i<br />
.<br />
α i f(x) n i <br />
f(x) ∈ F [x] n <br />
n E f(x) <br />
E[x] E F F <br />
E F [x] f(x) <br />
(f(x),f ′ (x)) = 1 <br />
f(x) F <br />
F 0 f(x) F p f(x) ≠ g(x p ) <br />
g(x) F [x] f(x) <br />
F =0 f ′ (x) < f(x) f(x) <br />
(f(x),f ′ (x)) ≠1 f ′ (x) <br />
F = p f ′ (x) <br />
f ′ (x) p <br />
f(x) =a 0 + a 1 x p + a 2 x 2p + ···+ a n x np <br />
F F (α) <br />
E F <br />
α ∈ E E = F (α) α <br />
<br />
<br />
Q( √ 3, √ 5)=Q( √ 3+ √ 5)<br />
<br />
Q( 3√ 5, √ 5 i) =Q( 6√ 5 i).<br />
<br />
<br />
i=1
E <br />
F α ∈ E E = F (α)<br />
F E<br />
F (α, β) <br />
f(x) g(x) <br />
α β K f(x) g(x) <br />
f(x) α = α 1 ,...,α n K g(x) β = β 1 ,...,β m K<br />
1 E F F <br />
a F <br />
a ≠ α i − α<br />
β − β j<br />
i j j ≠1 a(β − β j ) ≠ α i − α γ = α + aβ <br />
γ = α + aβ ≠ α i + aβ j ;<br />
γ − aβ j ≠ α i i, j j ≠1 h(x) ∈ F (γ)[x] h(x) =f(γ − ax)<br />
h(β) =f(α) =0 h(β j ) ≠0 j ≠1 h(x) g(x) <br />
F (γ)[x] β F (γ) <br />
β g(x) h(x) β ∈ F (γ) α = γ − aβ <br />
F (γ) F (α, β) =F (γ)<br />
<br />
<br />
<br />
G(E/F ) <br />
E F <br />
{σ i : i ∈ I} F <br />
F <br />
F {σi } = {a ∈ F : σ i (a) =a σ i }<br />
<br />
σ i (a) =a σ i (b) =b <br />
<br />
σ i (a ± b) =σ i (a) ± σ i (b) =a ± b<br />
σ i (ab) =σ i (a)σ i (b) =ab.<br />
a ≠0 σ i (a −1 )=[σ i (a)] −1 = a −1 σ i (0) = 0 σ i (1) = 1 σ i <br />
<br />
F G (F ) <br />
F <br />
F G = {α ∈ F : σ(α) =α σ ∈ G}<br />
F {σi } F {σ i } <br />
G (F ) F G <br />
σ : Q( √ 3, √ 5) → Q( √ 3, √ 5) √ 3<br />
− √ 3 Q( √ 5) Q( √ 3, √ 5) σ
E F <br />
E G(E/F ) = F <br />
<br />
G = G(E/F ) F ⊂ E G ⊂ E E E G<br />
G(E/F )=G(E/E G ) <br />
|G| =[E : E G ]=[E : F ].<br />
[E G : F ]=1 E G = F <br />
<br />
<br />
G E F = E G <br />
[E : F ] ≤|G|<br />
|G| = n n +1 α 1 ,...,α n+1 E <br />
F a i ∈ F <br />
a 1 α 1 + a 2 α 2 + ···+ a n+1 α n+1 =0.<br />
σ 1 = ,σ 2 ,...,σ n G <br />
<br />
σ 1 (α 1 )x 1 + σ 1 (α 2 )x 2 + ···+ σ 1 (α n+1 )x n+1 =0<br />
σ 2 (α 1 )x 1 + σ 2 (α 2 )x 2 + ···+ σ 2 (α n+1 )x n+1 =0<br />
<br />
σ n (α 1 )x 1 + σ n (α 2 )x 2 + ···+ σ n (α n+1 )x n+1 =0<br />
<br />
x i = a i i =1, 2,...,n+1 σ 1 <br />
<br />
a 1 α 1 + a 2 α 2 + ···+ a n+1 α n+1 =0.<br />
a i E F <br />
<br />
a i E F α i <br />
a 1 <br />
a 1 =1 <br />
α 2 ,...,α n+1 <br />
a 2 E F F E <br />
G σ i G σ i (a 2 ) ≠ a 2 σ i <br />
G <br />
x 1 = σ i (a 1 )=1 x 2 = σ i (a 2 ) ... x n+1 = σ i (a n+1 ) <br />
<br />
<br />
x 1 =1− 1=0<br />
x 2 = a 2 − σ i (a 2 )<br />
<br />
x n+1 = a n+1 − σ i (a n+1 )
σ i (a 2 ) ≠ a 2 <br />
<br />
<br />
a 1 ,...,a n+1 ∈ F <br />
E F F [x] <br />
E E E F <br />
F [x] E <br />
E[x]<br />
E F <br />
<br />
<br />
E F <br />
E F <br />
F = E G G E<br />
⇒ E F <br />
α E E = F (α) f(x) <br />
α F E f(x) <br />
F E f(x)<br />
⇒ E F <br />
E G(E/F ) = F |G(E/F )| =[E : F ] <br />
⇒ F = E G G E [E : F ] ≤<br />
|G| E F E F <br />
f(x) ∈ F [x] α E <br />
f(x) E[x] <br />
G f(x) E G α <br />
α 1 = α, α 2 ,...,α n E g(x) = ∏ n<br />
i=1 (x − α i) g(x) F<br />
g(α) =0 σ G g(x) <br />
σ g(x) g(x) <br />
g(x) F g(x) ≤ f(x) f(x) <br />
α f(x) =g(x)<br />
K F F = K G <br />
G K G = G(K/F )<br />
<br />
F = K G G G(K/F ) <br />
[K : F ] ≤|G| ≤|G(K/F )| =[K : F ].<br />
G = G(K/F ) <br />
<br />
<br />
Q( √ 3, √ 5) <br />
Q Q <br />
G(Q( √ 3, √ 5)/Q)
{,σ,τ,μ}<br />
Q( √ 3, √ 5)<br />
{,σ} {,τ} {,μ}<br />
Q( √ 3) Q( √ 5) Q( √ 15 )<br />
{}<br />
Q<br />
G(Q( √ 3, √ 5)/Q)<br />
<br />
F <br />
E F <br />
G(E/F ) <br />
K ↦→ G(E/K) K E F <br />
G(E/F )<br />
F ⊂ K ⊂ E <br />
[E : K] =|G(E/K)| [K : F ]=[G(E/F ):G(E/K)].<br />
F ⊂ K ⊂ L ⊂ E {} ⊂G(E/L) ⊂ G(E/K) ⊂ G(E/F )<br />
K F G(E/K) G(E/F )<br />
<br />
G(K/F ) ∼ = G(E/F )/G(E/K).<br />
G(E/K) =G(E/L) =G K L <br />
G K = L K ↦→ G(E/K) <br />
G G(E/F ) K G <br />
F ⊂ K ⊂ E E K G(E/K) =G <br />
K ↦→ G(E/K) <br />
|G(E/K)| =[E : K] <br />
|G(E/F )| =[G(E/F ):G(E/K)] ·|G(E/K)| =[E : F ]=[E : K][K : F ].<br />
[K : F ]=[G(E/F ):G(E/K)]<br />
<br />
<br />
K F σ <br />
G(E/F ) τ G(E/K) σ −1 τσ G(E/K) <br />
σ −1 τσ(α) =α α ∈ K f(x) <br />
α F σ(α) f(x) K K F <br />
τ(σ(α)) = σ(α) σ −1 τσ(α) =α<br />
G(E/K) G(E/F ) <br />
F = K G(K/F ) τ ∈ G(E/K) σ ∈ G(E/F ) τ ∈ G(E/K) <br />
τσ = στ α ∈ K<br />
τ(σ(α)) = σ(τ(α)) = σ(α);
σ(α) G(E/K) σ σ K <br />
σ K F σ(α) ∈ K α ∈ K σ ∈ G(K/F )<br />
G(K/F ) F β K <br />
G(K/F ) σ(β) =β σ ∈ G(E/F )<br />
β F G(E/F )<br />
K F <br />
G(K/F ) ∼ = G(E/F )/G(E/K).<br />
σ ∈ G(E/F ) σ K K σ K K<br />
σ K ∈ G(K/F )<br />
φ : G(E/F ) → G(K/F ) σ ↦→ σ K <br />
<br />
φ G(E/K) <br />
φ(στ) =(στ) K = σ K τ K = φ(σ)φ(τ).<br />
|G(E/F )|/|G(E/K)| =[K : F ]=|G(K/F )|.<br />
φ G(K/F ) φ <br />
<br />
G(K/F ) ∼ = G(E/F )/G(E/K).<br />
E<br />
{}<br />
L<br />
G(E/L)<br />
K<br />
G(E/K)<br />
F G(E/F )<br />
G(E/F ) E<br />
<br />
f(x) =x 4 − 2 <br />
Q <br />
x 4 − 2 f(x) Q( 4√ 2,i) <br />
f(x) (x 2 + √ 2)(x 2 − √ 2) f(x) ± 4√ 2 ± 4√ 2 i <br />
4√ 2 Q i x 2 +1 Q( 4√ 2) <br />
f(x) Q( 4√ 2)(i) =Q( 4√ 2,i)<br />
[Q( 4√ 2) : Q] =4 i Q( 4√ 2) [Q( 4√ 2,i):<br />
Q( 4√ 2)]=2 [Q( 4√ 2,i):Q] =8 <br />
{1, 4√ 2, ( 4√ 2) 2 , ( 4√ 2) 3 ,i,i 4√ 2,i( 4√ 2) 2 ,i( 4√ 2) 3 }<br />
Q( 4√ 2,i) Q Q Q( 4√ 2,i) <br />
<br />
G f(x) 8 σ <br />
σ( 4√ 2) = i 4√ 2 σ(i) =i τ
τ(i) =−i G 4 2 <br />
G {,σ,σ 2 ,σ 3 ,τ,στ,σ 2 τ,σ 3 τ}<br />
τ 2 = σ 4 = τστ = σ −1 G <br />
D 4 G <br />
Q( 4√ 2,i)<br />
Q( 4√ 2) Q( 4√ 2 i) Q( √ 2,i) Q((1 + i) 4√ 2)Q((1 − i) 4√ 2)<br />
Q( √ 2) Q(i) Q( √ 2 i)<br />
Q<br />
<br />
D 4<br />
{,σ 2 ,τ,σ 2 τ}{,σ,σ 2 ,σ 3 }{,σ 2 ,στ,σ 3 τ}<br />
{,τ} {,σ 2 τ} {,σ 2 } {,στ} {,σ 3 τ}<br />
{}<br />
<br />
x 4 − 2
f(x) <br />
f(x) <br />
n <br />
ax 2 + bx + c =0 <br />
x = −b ± √ b 2 − 4ac<br />
.<br />
2a<br />
n <br />
<br />
E F <br />
<br />
F = F 0 ⊂ F 1 ⊂ F 2 ⊂···⊂F r = E<br />
i =1, 2,...,rF i = F i−1 (α i ) α n i<br />
i<br />
∈ F i−1 n i <br />
f(x) F K f(x) F<br />
F <br />
f(x) <br />
f(x)<br />
x n − a <br />
x n − 1 <br />
x n − 1 n <br />
<br />
x n − 1 Q <br />
1,ω,ω 2 ,...,ω n−1 <br />
( ) ( )<br />
2π 2π<br />
ω = + i .<br />
n n<br />
x n − 1 Q Q(ω)<br />
<br />
G <br />
G = H n ⊃ H n−1 ⊃···⊃H 1 ⊃ H 0 = {e},<br />
H i H i+1 G {H i } <br />
H i+1 /H i <br />
{} ⊂A 3 ⊂ S 3 S 3 S 5
F E x n − a<br />
F a ∈ F G(E/F ) <br />
x n − a n√ a, ω n√ a,...,ω n−1 n√ a ω n <br />
F n ζ <br />
x n − a x n − a ζ,ωζ,...,ω n−1 ζ E = F (ζ) G(E/F )<br />
x n − a G(E/F ) <br />
σ τ G(E/F ) σ(ζ) =ω i ζ τ(ζ) =ω j ζF<br />
<br />
στ(ζ) =σ(ω j ζ)=ω j σ(ζ) =ω i+j ζ = ω i τ(ζ) =τ(ω i ζ)=τσ(ζ).<br />
στ = τσ G(E/F ) G(E/F ) <br />
F n ω <br />
n α x n − a α ωα<br />
x n − a ω =(ωα)/α E K = F (ω) <br />
F ⊂ K ⊂ E K x n − 1 K F <br />
σ G(F (ω)/F ) σ(ω) <br />
σ(ω) =ω i i x n −1 ω τ(ω) =ω j<br />
G(F (ω)/F ) <br />
στ(ω) =σ(ω j )=[σ(ω)] j = ω ij =[τ(ω)] i = τ(ω i )=τσ(ω).<br />
G(F (ω)/F ) <br />
{} ⊂G(E/F (ω)) ⊂ G(E/F )<br />
G(E/F (ω)) <br />
G(E/F ) <br />
G(E/F )/G(E/F (ω)) ∼ = G(F (ω)/F )<br />
F <br />
F = F 0 ⊂ F 1 ⊂ F 2 ⊂···⊂F r = E<br />
F <br />
F = K 0 ⊂ K 1 ⊂ K 2 ⊂···⊂K r = K<br />
K E K i K i−1 <br />
<br />
E F <br />
F = F 0 ⊂ F 1 ⊂ F 2 ⊂···⊂F r = E<br />
i =1, 2,...,rF i = F i−1 (α i ) α n i<br />
i<br />
F <br />
∈ F i−1 n i <br />
F = K 0 ⊂ K 1 ⊂ K 2 ⊂···⊂K r = K<br />
K ⊇ E K 1 x n 1<br />
− α n 1<br />
1 <br />
α 1 ,α 1 ω, α 1 ω 2 ,...,α 1 ω n1−1 ω n 1 F<br />
n 1 K 1 = F (α ! ) <br />
F n 1 β x n 1<br />
− α n 1<br />
1
x n 1<br />
− α n 1<br />
1 β,ωβ,...,ω n 1−1 ω n 1 <br />
K 1 = F (ωβ) K 1 F F 1 <br />
<br />
F = K 0 ⊂ K 1 ⊂ K 2 ⊂···⊂K r = K<br />
K i K i−1 K i ⊇ F i i =1, 2,...,r<br />
<br />
f(x) F [x] F =0 f(x) <br />
f(x) F <br />
f(x) E F <br />
F = F 0 ⊂ F 1 ⊂···⊂F n = E E <br />
f(x) F i F i−1 G(E/F i )<br />
G(E/F i−1 ) <br />
G(E/F )<br />
{} ⊂G(E/F n−1 ) ⊂···⊂G(E/F 1 ) ⊂ G(E/F ).<br />
<br />
G(E/F i−1 )/G(E/F i ) ∼ = G(F i /F i−1 ).<br />
G(F i /F i−1 ) G(E/F ) <br />
<br />
<br />
<br />
<br />
S 5 <br />
<br />
p S p <br />
p S p <br />
G S p σ τ <br />
p σ = (12) τ p τ n <br />
p 1 ≤ n
f(x) =x 5 − 6x 3 − 27x − 3<br />
f(x) =x 5 − 6x 3 − 27x − 3 ∈ Q[x] <br />
f(x) Q S 5 f(x) <br />
f(x) f ′ (x) =5x 4 −18x 2 −<br />
27 f ′ (x) =0 f ′ (x) <br />
√<br />
6 √ 6+9<br />
x = ± .<br />
5<br />
f(x) <br />
f(x) −3 −2 −2 0 0 <br />
4 f(x) <br />
f(x) K f(x) f(x)<br />
K K Q <br />
f(x) G(K/Q) S 5 f <br />
σ ∈ G(K/Q) σ(a) =b a b<br />
f(x) C a + bi ↦→ a − bi <br />
G(K/Q) α <br />
f(x) [Q(α) :Q] =5 Q(α) <br />
K [K : Q] [K : Q] =|G(K/Q)| <br />
G(K/Q) ⊂ S 5 G(K/Q) 5 S 5 <br />
5 G(K/Q)
S 5 S 5 f(x) <br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
C[x] C<br />
E <br />
<br />
α ∈ E E = C(α) α f(x) C[x] <br />
L f(x) C E <br />
L C<br />
L C L f(x)(x 2 +1)<br />
R L R K <br />
G G(L/R) L ⊃ K ⊃ R |G(L/K)| =[L : K] [L : R] =[L :<br />
K][K : R] [K : R] K = R(β) β <br />
f(x) K = R<br />
G(L/R) G(L/C) 2 <br />
L ≠ C |G(L/C)| ≥2 <br />
G G(L/C) <br />
E G [E : C] =2 γ ∈ E<br />
x 2 + bx + c C[x] (−b ± √ b 2 − 4c )/2<br />
C b 2 − 4c C L = C<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Q
G(Q( √ 30 )/Q)<br />
G(Q( 4√ 5)/Q)<br />
G(Q( √ 2, √ 3, √ 5)/Q)<br />
G(Q( √ 2, 3√ 2,i)/Q)<br />
G(Q( √ 6,i)/Q)<br />
<br />
<br />
x 3 +2x 2 − x − 2 Q<br />
x 4 +2x 2 +1 Q<br />
x 4 + x 2 +1 Z 3<br />
x 3 + x 2 +1 Z 2<br />
(729) (9)<br />
Q[x] <br />
<br />
x 5 − 12x 2 +2<br />
x 5 − 4x 4 +2x +2<br />
x 3 − 5<br />
x 4 − x 2 − 6<br />
x 5 +1<br />
(x 2 − 2)(x 2 +2)<br />
x 8 − 1<br />
x 8 +1<br />
x 4 − 3x 2 − 10<br />
<br />
Q[x]<br />
x 4 − 1<br />
x 4 − 8x 2 +15<br />
x 4 − 2x 2 − 15<br />
x 3 − 2<br />
<br />
Z 2 <br />
S 3 <br />
Z 3 <br />
F ⊂ K ⊂ E E F E <br />
K<br />
G n |G| n!<br />
F ⊂ E f(x) F f(x) E<br />
f(x) Q[x] 7 <br />
p f(x) ∈ Q[x] p <br />
S p p p ≥ 5 <br />
p <br />
p Z p (t) Z p <br />
f(x) =x p − t Z p (t)[x] f(x) <br />
E F K L <br />
σ ∈ G(E/F ) σ(K) =L K L <br />
K L G(E/K) G(E/L)<br />
G(E/F )<br />
σ ∈ (R) a σ(a) > 0
K x 3 + x 2 +1∈ Z 2 [x] K <br />
<br />
F (F ) ≠2 f(x) =<br />
ax 2 + bx + c F ( √ α ) α = b 2 − 4ac<br />
<br />
<br />
K F E F<br />
K [E : F ]=2 E F [x]<br />
<br />
Φ p (x) = xp − 1<br />
x − 1 = xp−1 + x p−2 + ···+ x +1<br />
Q p ω Φ p (x) <br />
Q(ω)<br />
ω, ω 2 ,...,ω p−1 Φ p (x) <br />
Φ p (x)<br />
G(Q(ω)/Q) p − 1<br />
G(Q(ω)/Q) Q<br />
F E <br />
F G(E/F ) F ⊂ K ⊂ L ⊂ E <br />
{} ⊂G(E/L) ⊂ G(E/K) ⊂ G(E/F )<br />
F f(x) ∈ F [x] <br />
n E f(x) α 1 ,...,α n f(x) E <br />
Δ= ∏ i
A
B<br />
<br />
<br />
<br />
<br />
A ∩ B = {2} B ∩ C = {5}<br />
A × B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c, 1), (c, 2), (c, 3)} <br />
A × D = ∅<br />
x ∈ A ∪ (B ∩ C) x ∈ A x ∈ B ∩ C x ∈ A ∪ B<br />
A ∪ C x ∈ (A ∪ B) ∩ (A ∪ C) A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C)<br />
x ∈ (A ∪ B) ∩ (A ∪ C) x ∈ A ∪ B A ∪ C x ∈ A x <br />
B C x ∈ A ∪ (B ∩ C) (A ∪ B) ∩ (A ∪ C) ⊂ A ∪ (B ∩ C) <br />
A ∪ (B ∩ C) =(A ∪ B) ∩ (A ∪ C)<br />
(A ∩ B) ∪ (A \ B) ∪ (B \ A) =(A ∩ B) ∪ (A ∩ B ′ ) ∪ (B ∩ A ′ )=[A ∩ (B ∪<br />
B ′ )] ∪ (B ∩ A ′ )=A ∪ (B ∩ A ′ )=(A ∪ B) ∩ (A ∪ A ′ )=A ∪ B<br />
A \ (B ∪ C) =A ∩ (B ∪ C) ′ =(A ∩ A) ∩ (B ′ ∩ C ′ )=(A ∩ B ′ ) ∩ (A ∩ C ′ )=<br />
(A \ B) ∩ (A \ C)<br />
f(2/3) <br />
f(1/2) = 3/4 f(2/4) = 3/8 <br />
f f(R) ={x ∈ R : x>0} f <br />
f(R) ={x : −1 ≤ x ≤ 1}<br />
f(n) =n +1<br />
x, y ∈ A g(f(x)) = (g◦f)(x) =(g◦f)(y) =g(f(y)) <br />
f(x) =f(y) x = y g ◦ f c ∈ C c =(g ◦ f)(x) =g(f(x))<br />
x ∈ A f(x) ∈ B g <br />
f −1 (x) =(x +1)/(x − 1)<br />
y ∈ f(A 1 ∪ A 2 ) x ∈ A 1 ∪ A 2 <br />
f(x) =y y ∈ f(A 1 ) f(A 2 ) y ∈ f(A 1 ) ∪ f(A 2 ) <br />
f(A 1 ∪ A 2 ) ⊂ f(A 1 ) ∪ f(A 2 ) y ∈ f(A 1 ) ∪ f(A 2 ) y ∈ f(A 1 ) f(A 2 )<br />
x A 1 A 2 f(x) =y x ∈ A 1 ∪ A 2<br />
f(x) =y f(A 1 )∪f(A 2 ) ⊂ f(A 1 ∪A 2 ) f(A 1 ∪A 2 )=f(A 1 )∪f(A 2 )<br />
<br />
0 <br />
X = N ∪{ √ 2 } x ∼ y x + y ∈ N
S(1) : [1(1 + 1)(2(1) + 1)]/6 = 1 = 1 2 <br />
S(k) :1 2 +2 2 + ···+ k 2 =[k(k + 1)(2k +1)]/6 <br />
1 2 +2 2 + ···+ k 2 +(k +1) 2 =[k(k + 1)(2k +1)]/6+(k +1) 2<br />
=[(k + 1)((k + 1) + 1)(2(k + 1) + 1)]/6,<br />
S(k +1) S(n) n<br />
S(4):4!=24> 16 = 2 4 S(k) :k! > 2 k <br />
(k +1)!=k!(k +1)> 2 k · 2=2 k+1 S(k +1) S(n) <br />
n<br />
<br />
S(0) : (1 + x) 0 − 1=0≥ 0=0· x <br />
S(k) :(1+x) k − 1 ≥ kx <br />
(1 + x) k+1 − 1=(1+x)(1 + x) k − 1<br />
=(1+x) k + x(1 + x) k − 1<br />
≥ kx + x(1 + x) k<br />
≥ kx + x<br />
=(k +1)x,<br />
S(k +1) S(n) n<br />
<br />
f 1 =1 f 2 =1 f n+2 = f n+1 + f n <br />
<br />
<br />
<br />
(a, b) =1 r s ar + bs =1<br />
acr + bcs = c<br />
2 3 6n +16n +5 <br />
6k +5<br />
<br />
<br />
3+7Z = {...,−4, 3, 10,...} 18 + 26Z 5+6Z<br />
<br />
<br />
· 1 5 7 11<br />
1 1 5 7 11<br />
5 5 1 11 7<br />
7 7 11 1 5<br />
11 11 7 5 1
( 1 2 ···<br />
σ =<br />
a 1 a 2 ···<br />
) n<br />
a n<br />
S n a i n a 1 n − 1 <br />
a 2 ... a n−1 a n <br />
σ n(n − 1) ···2 · 1=n! <br />
<br />
(aba −1 ) n =(aba −1 )(aba −1 ) ···(aba −1 )<br />
= ab(aa −1 )b(aa −1 )b ···b(aa −1 )ba −1<br />
= ab n a −1 .<br />
abab =(ab) 2 = e = a 2 b 2 = aabb ba = ab<br />
H 1 = {} H 2 = {,ρ 1 ,ρ 2 } H 3 = {,μ 1 } H 4 = {,μ 2 } H 5 =<br />
{,μ 3 } S 3 <br />
G 1=1+0 √ 2 (a + b √ 2)(c + d √ 2) = (ac +<br />
2bd)+(ad + bc) √ 2 G (a + b √ 2) −1 = a/(a 2 − 2b 2 ) −<br />
b √ 2/(a 2 − 2b 2 )<br />
S 3 <br />
ba = a 4 b = a 3 ab = ab<br />
<br />
<br />
<br />
12 10<br />
7Z = {...,−7, 0, 7, 14,...} {0, 3, 6, 9, 12, 15, 18, 21} {0} {0, 6}<br />
{0, 4, 8} {0, 3, 6, 9} {0, 2, 4, 6, 8, 10} {1, 3, 7, 9} {1, −1,i,−i}<br />
<br />
( ) ( )<br />
1 0 −1 0<br />
,<br />
,<br />
0 1 0 −1<br />
( ) 0 −1<br />
,<br />
1 0<br />
( ) 0 1<br />
.<br />
−1 0<br />
<br />
( ) 1 0<br />
,<br />
0 1<br />
( ) 1 −1<br />
,<br />
1 0<br />
( ) −1 1<br />
,<br />
−1 0<br />
( ) 0 1<br />
,<br />
−1 1<br />
( ) ( )<br />
0 −1 −1 0<br />
,<br />
.<br />
1 −1 0 −1<br />
0 1, −1<br />
1, 2, 3, 4, 6, 8, 12, 24<br />
−3+3i 43 − 18i i<br />
√ 3+i −3
√ 2 (7π/4) 2 √ 2 (π/4) 3 (3π/2)<br />
(1 − i)/2 16(i − √ 3) −1/4<br />
292 1523<br />
|〈g〉∩〈h〉| =1<br />
g, h ∈ G <br />
m n (g −1 ) m = e (gh) mn = e <br />
G G<br />
g G g G<br />
〈g〉 G<br />
<br />
<br />
<br />
(12453) (13)(25)<br />
(135)(24) (14)(23) (1324) (134)(25) (17352)<br />
(16)(15)(13)(14) (16)(14)(12)<br />
(a 1 ,a 2 ,...,a n ) −1 =(a 1 ,a n ,a n−1 ,...,a 2 )<br />
{(13), (13)(24), (132), (134), (1324), (1342)} <br />
(12345)(678)<br />
<br />
(1), (a 1 ,a 2 )(a 3 ,a 4 ), (a 1 ,a 2 ,a 3 ), (a 1 ,a 2 ,a 3 ,a 4 ,a 5 )<br />
A 5 <br />
(123)(12) (12)(123)<br />
(ab)(bc) (ab)(cd)<br />
στσ −1 (σ(a i )) = σ(a i+1 )<br />
<br />
<br />
g h G<br />
60<br />
<br />
<br />
〈8〉 1+〈8〉 2+〈8〉 3+〈8〉 4+〈8〉 5+〈8〉 6+〈8〉 7+〈8〉 <br />
3Z 1+3Z 2+3Z<br />
4 φ(15) ≡ 4 8 ≡ 1( 15)<br />
g 1 ∈ gH g 1 ∈ Hg gH ⊂ Hg<br />
g(H ∩ K) =gH ∩ gK<br />
(m, n) =1 φ(mn) =φ(m)φ(n)
26! − 1<br />
2791 11213525032442<br />
31 14<br />
n =11· 41 n = 8779 · 4327<br />
<br />
<br />
(0000) /∈ C<br />
2 2<br />
3 4<br />
d =2 d =1<br />
<br />
(00000), (00101), (10011), (10110)<br />
⎛ ⎞<br />
0 1<br />
0 0<br />
G =<br />
1 0<br />
⎜ ⎟<br />
⎝0 1⎠<br />
1 1<br />
(000000), (010111), (101101), (111010)<br />
⎛ ⎞<br />
1 0<br />
0 1<br />
1 0<br />
G =<br />
1 1<br />
⎜ ⎟<br />
⎝0 1⎠<br />
1 1<br />
<br />
<br />
⎛ ⎞<br />
1<br />
1<br />
G =<br />
0<br />
.<br />
⎜ ⎟<br />
⎝0⎠<br />
1
⎛ ⎞<br />
1 0<br />
0 1<br />
G = ⎜ ⎟<br />
⎝1 1⎠ .<br />
1 0<br />
<br />
C (10000) + C (01000) + C (00100) + C (00010) + C (11000) + C<br />
(01100) + C (01010) + C C <br />
<br />
∈ C <br />
↦→ + <br />
20 <br />
<br />
<br />
<br />
Z <br />
φ : C ∗ → GL 2 (R) <br />
( ) a b<br />
φ(a + bi) = .<br />
−b a<br />
<br />
Z n n k ↦→ (2kπ/n)<br />
Q <br />
<br />
<br />
12 5<br />
<br />
<br />
<br />
a G φ : G → H <br />
φ(a) H<br />
Z 6 Z 6 <br />
φ g 1 = h 1 k 1 g 2 = h 2 k 2 <br />
φ(g 1 )=φ(g 2 )
A 4 (12)A 4<br />
A 4 A 4 (12)A 4<br />
(12)A 4 (12)A 4 A 4<br />
D 4 S 4 <br />
a ∈ G G aH G/H<br />
g ∈ G i g : G → G i g : x ↦→ gxg −1<br />
G i g (H)<br />
〈g〉 G y <br />
G x ∈ C(g) yxy −1 C(g) (yxy −1 )g = g(yxy −1 )<br />
g ∈ G h ∈ G ′ h = aba −1 b −1 <br />
ghg −1 = gaba −1 b −1 g −1<br />
=(gag −1 )(gbg −1 )(ga −1 g −1 )(gb −1 g −1 )<br />
=(gag −1 )(gbg −1 )(gag −1 ) −1 (gbg −1 ) −1 .<br />
h = h 1 ···h n h i = a i b i a −1<br />
i<br />
b −1<br />
i<br />
ghg −1 <br />
ghg −1 = gh 1 ···h n g −1 =(gh 1 g −1 )(gh 2 g −1 ) ···(gh n g −1 )<br />
<br />
<br />
{1} <br />
φ(m + n) =7(m + n) =7m +7n = φ(m)+φ(n) φ <br />
<br />
φ : Z 24 → Z 18 φ <br />
Z 24 φ Z 18 <br />
<br />
a, b ∈ G φ(a)φ(b) =φ(ab) =φ(ba) =φ(b)φ(a)<br />
<br />
<br />
<br />
<br />
<br />
1 [<br />
‖ + ‖ 2 + ‖‖ 2 −‖‖ 2] = 1 [<br />
〈x + y, x + y〉−‖‖ 2 −‖‖ 2]<br />
2<br />
2<br />
= 1 [<br />
‖‖ 2 +2〈x, y〉 + ‖‖ 2 −‖‖ 2 −‖‖ 2]<br />
2<br />
= 〈, 〉.
SO(2) O(3)<br />
〈, 〉 = 〈, 〉<br />
<br />
( ) 5 2<br />
.<br />
2 1<br />
: O(n) → R ∗ SO(n)<br />
<br />
p6m<br />
<br />
<br />
<br />
{0} ⊂〈6〉 ⊂〈3〉 ⊂Z 12 {(1)}×{0} ⊂{(1), (123), (132)}×{0} ⊂<br />
S 3 ×{0} ⊂S 3 ×〈2〉 ⊂S 3 × Z 4 <br />
<br />
N G/N <br />
N = N n ⊃ N n−1 ⊃···⊃N 1 ⊃ N 0 = {e}<br />
G/N = G n /N ⊃ G n−1 /N ⊃···G 1 /N ⊃ G 0 /N = {N}.<br />
D n 2<br />
G/G ′ <br />
<br />
<br />
0 R 2 \{0} X = {1, 2, 3, 4}<br />
X (1) = {1, 2, 3} X (12) = {3} X (13) = {2} X (23) = {1} X (123) =<br />
X (132) = ∅ G 1 = {(1), (23)} G 2 = {(1), (13)} G 3 = {(1), (12)}<br />
O 1 = O 2 = O 3 = {1, 2, 3}<br />
S 4 <br />
O (1) = {(1)},<br />
O (12) = {(12), (13), (14), (23), (24), (34)},<br />
O (12)(34) = {(12)(34), (13)(24), (14)(23)},<br />
O (123) = {(123), (132), (124), (142), (134), (143), (234), (243)},<br />
O (1234) = {(1234), (1243), (1324), (1342), (1423), (1432)}.<br />
1+3+6+6+8=24<br />
(3 4 +3 1 +3 2 +3 1 +3 2 +3 2 +3 3 +3 3 )/8 = 21
S 4 <br />
(abcd) <br />
(ab)(cd) <br />
(ab)(cd)(ef) <br />
(abc)(def) <br />
<br />
(1 · 2 6 +3· 2 4 +4· 2 3 +2· 2 2 +2· 2 1 )/12 = 13<br />
(1 · 2 8 +3· 2 6 +2· 2 4 )/6 = 80<br />
x ∈ gC(a)g −1 g −1 xg ∈ C(a)<br />
<br />
<br />
|G| =18=2· 3 2 2 2 <br />
3 9<br />
3 S 4 P 1 = {(1), (123), (132)} P 2 =<br />
{(1), (124), (142)} P 3 = {(1), (134), (143)} P 4 = {(1), (234), (243)}<br />
|G| =96=2 5 · 3 G 2 <br />
<br />
2 H K |H ∩ K| ≥16 HK<br />
(32 · 32)/8 = 128 H ∩ K <br />
H K 2 <br />
G p p 2 <br />
q q 2 <br />
<br />
G G |G| =3· 5 · 17 <br />
<br />
N(H) G <br />
H G N(H)g ↦→ g −1 Hg <br />
aG ′ ,bG ′ ∈ G/G ′ (aG ′ )(bG ′ )=abG ′ = ab(b −1 a −1 ba)G ′ =<br />
(abb −1 a −1 )baG ′ = baG ′ <br />
<br />
<br />
7Z Q( √ 2) R <br />
{1, 3, 7, 9} {1, 2, 3, 4, 5, 6} <br />
{( ) 1 0<br />
,<br />
0 1<br />
( ) 1 1<br />
,<br />
0 1<br />
( ) 1 0<br />
,<br />
1 1<br />
( ) 0 1<br />
,<br />
1 0<br />
( ) 1 1<br />
,<br />
1 0<br />
( ) } 0 1<br />
, .<br />
1 1<br />
{0} {0, 9} {0, 6, 12} {0, 3, 6, 9, 12, 15} {0, 2, 4, 6, 8, 10, 12, 14, 16}<br />
<br />
φ : C → R φ(i) =a
φ( √ 2)=a<br />
φ : Q( √ 2) → Q( √ 3) <br />
x ≡ 17 ( 55) x ≡ 214 ( 2772)<br />
I ≠ {0} 1 ∈ I<br />
φ(a)φ(b) =φ(ab) =φ(ba) =φ(b)φ(a)<br />
a ∈ R a ≠0 a R<br />
b ∈ R ab =1<br />
(a + b) 2 (−ab) 2 <br />
a/b, c/d ∈ Z (p) a/b + c/d =(ad + bc)/bd (a/b) · (c/d) =<br />
(ac)/(bd) Z (p) (bd, p) =1<br />
x 2 = x x ≠0 R x =1<br />
M 2 (R)<br />
<br />
<br />
9x 2 +2x +5 8x 4 +7x 3 +2x 2 +7x<br />
5x 3 +6x 2 − 3x +4 = (5x 2 +2x +1)(x − 2) + 6 4x 5 − x 3 + x 2 +4 =<br />
(4x 2 +4)(x 3 +3)+4x 2 +2<br />
Z 12 3 4<br />
(2x +1)<br />
<br />
x 2 + x +8=(x +2)(x +9)<br />
Z <br />
<br />
φ : R → S φ : R[x] → S[x] <br />
φ(a 0 + a 1 x + ···+ a n x n )=φ(a 0 )+φ(a 1 )x + ···+ φ(a n )x n <br />
<br />
Φ n (x) = xn − 1<br />
x − 1 = xn−1 + x n−2 + ···+ x +1<br />
Φ p (x) Q <br />
p<br />
F [x]<br />
<br />
<br />
z −1 = 1/(a + b √ 3 i) =(a − b √ 3 i)/(a 2 +3b 2 ) Z[ √ 3 i] <br />
a 2 +3b 2 =1 a = ±1,b=0<br />
5=−i(1 + 2i)(2 + i) 6+8i = −i(1 + i) 2 (2 + i) 2 <br />
<br />
z = a + bi w = c + di ≠0 Z[i] z/w ∈ Q(i)
a = ub u ν(b) ≤ ν(ub) ≤ ν(a) <br />
ν(a) ≤ ν(b)<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
(a ∨ b ∨ a ′ ) ∧ a<br />
a<br />
b<br />
a<br />
a ′<br />
a ∨ (a ∧ b)<br />
a<br />
b<br />
a<br />
<br />
a ′ ∧ [(a ∧ b ′ ) ∨ b] =a ∧ (a ∨ b)
I,J R I + J = {r + s :<br />
r ∈ I s ∈ J} R I J r 1 ,r 2 ∈ I <br />
s 1 ,s 2 ∈ J (r 1 + s 1 )+(r 2 + s 2 )=(r 1 + r 2 )+(s 1 + s 2 ) I + J a ∈ R<br />
a(r 1 + s 1 )=ar 1 + as 1 ∈ I + J I + J R<br />
<br />
(⇒) a = b ⇒ (a ∧ b ′ ) ∨ (a ′ ∧ b) =(a ∧ a ′ ) ∨ (a ′ ∧ a) =O ∨ O = O<br />
(⇐) (a ∧ b ′ ) ∨ (a ′ ∧ b) =O ⇒ a ∨ b =(a ∨ a) ∨ b = a ∨ (a ∨ b) =a ∨ [I ∧ (a ∨ b)] =<br />
a ∨ [(a ∨ a ′ ) ∧ (a ∨ b)] = [a ∨ (a ∧ b ′ )] ∨ [a ∨ (a ′ ∧ b)] = a ∨ [(a ∧ b ′ ) ∨ (a ′ ∧ b)] = a ∨ 0=a <br />
a ∨ b = b<br />
<br />
<br />
<br />
Q( √ 2, √ 3) {1, √ 2, √ 3, √ 6 } Q<br />
{1,x,x 2 ,...,x n−1 } P n <br />
2 {(1, 0, −3), (0, 1, 2)} <br />
<br />
0=α0 =α(−v + v) =α(−v)+αv −αv = α(−v)<br />
v 0 =0,v 1 ,...,v n ∈ V α 0 ≠0,α 1 ,...,α n ∈ F α 0 v 0 +<br />
···+ α n v n =0<br />
u, v ∈ (T ) α ∈ F <br />
T (u + v) =T (u)+T (v) =0<br />
T (αv) =αT (v) =α0 =0.<br />
u + v, αv ∈ (T ) (T ) V <br />
T (u) =T (v) T (u − v) =T (u) − T (v) =0 <br />
u − v =0 u = v<br />
u, u ′ ∈ U v, v ′ ∈ V <br />
(u + v)+(u ′ + v ′ )=(u + u ′ )+(v + v ′ ) ∈ U + V<br />
α(u + v) =αu + αv ∈ U + V.<br />
<br />
<br />
x 4 − (2/3)x 2 − 62/9 x 4 − 2x 2 +25<br />
{1, √ 2, √ 3, √ 6 } {1,i, √ 2, √ 2 i} {1, 2 1/6 , 2 1/3 , 2 1/2 , 2 2/3 , 2 5/6 }<br />
Q( √ 3, √ 7)<br />
Z 2 [x]/〈x 3 + x +1〉 α 1+α<br />
α 2 1+α 2 α + α 2 1+α + α 2 α 3 + α +1=0<br />
<br />
E F K E <br />
α ∈ K α F α
E p(x) =β 0 + β 1 x + ···+ β n x n <br />
E[x] α F (β 0 ,...,β n )<br />
{1, √ 3, √ 7, √ 21 } Q( √ 3, √ 7) Q Q( √ 3, √ 7)⊃<br />
Q( √ 3+ √ 7) [Q( √ 3, √ 7) : Q] =4 [Q( √ 3+ √ 7) : Q] =2 <br />
√ 3+ √ 7 Q( √ 3, √ 7)=Q( √ 3+ √ 7)<br />
β ∈ F (α) F β = p(α)/q(α) p q <br />
α q(α) ≠0 F β F <br />
f(x) ∈ F [x] f(β) =0 f(x) =a 0 + a 1 x + ···+ a n x n <br />
<br />
( ) ( ) ( ) p(α)<br />
p(α)<br />
p(α) n<br />
0=f(β) =f = a 0 + a 1 + ···+ a n .<br />
q(α)<br />
q(α)<br />
q(α)<br />
q(α) n F [x] α <br />
<br />
<br />
<br />
<br />
<br />
Z 2 (α) x 3 + x 2 +1<br />
α <br />
p(x) Z 3 [x] 3 <br />
Z 3 [x]/〈p(x)〉 27 <br />
x 5 − 1=(x +1)(x 4 + x 3 + x 2 + x +1) x 9 − 1=(x +1)(x 2 + x +<br />
1)(x 6 + x 3 +1)<br />
<br />
x 7 − 1=(x +1)(x 3 + x +1)(x 3 + x 2 +1)<br />
<br />
p(x) ∈ F [x] p(x) ∈ E[x]<br />
α F n <br />
β ∈ F (α) β = a 0 + a 1 α + ···+ a n−1 α n−1 a i ∈ F q n <br />
n (a 0 ,a 1 ,...,a n−1 )<br />
x p−1 − 1 Z p <br />
<br />
<br />
Z 2 Z 2 × Z 2 × Z 2 <br />
Q x 3 +2x 2 − x − 2=(x − 1)(x +1)(x +2) <br />
Z 3 x 4 + x 2 +1=(x +1) 2 (x +2) 2 <br />
<br />
[(729) : (9)] = [(729) : (3)]/[(9) : (3)] = 6/2 = 3,
G((729)/ (9)) ∼ = Z 3 G((729)/ (9)) σ σ 3 6(α) =<br />
α 36 = α 729 α ∈ (729)<br />
S 5 S 3 <br />
Q(i)<br />
E F [x] <br />
[E : F ] 6 3 G(E/F ) S 3<br />
3 Z 3 S 3 <br />
G S n <br />
<br />
<br />
ω, ω 2 ,...,ω p−1 ω ≠1 ω i Φ p <br />
Φ p (ω i )<br />
ω ω, ω 2 ,...,ω p−1 φ i : Q(ω) → Q(ω i ) <br />
φ i (a 0 + a 1 ω + ···+ a p−2 ω p−2 )=a 0 + a 1 ω i + ···+ c p−2 (ω i ) p−2 ,<br />
a i ∈ Q φ i φ 2 <br />
G(Q(ω)/Q)<br />
{ω, ω 2 ,...,ω p−1 } Q(ω) Q <br />
ω, ω 2 ,...,ω p−1 G(Q(ω)/Q)
C<br />
<br />
<br />
<br />
<br />
a ∈ A a A <br />
N <br />
Z <br />
Q <br />
R <br />
C <br />
A ⊂ B A B <br />
∅ <br />
A ∪ B A B <br />
A ∩ B A B <br />
A ′ A <br />
A \ B A B <br />
A × B A B <br />
A n A ×···×A n <br />
id <br />
f −1 f <br />
a ≡ b ( n) a b n <br />
n! n <br />
( n<br />
k)<br />
n!/(k!(n − k)!)<br />
<br />
a | b a b <br />
(a, b) a b <br />
P(X) X <br />
(m, n) m n <br />
Z n n <br />
U(n) Z n <br />
M n (R) n × n R <br />
A A <br />
GL n (R) <br />
Q 8 <br />
C ∗ <br />
|G|
R ∗ <br />
Q ∗ <br />
SL n (R) <br />
Z(G) <br />
〈a〉 a <br />
|a| a <br />
θ θ + i θ <br />
T <br />
S n n <br />
(a 1 ,a 2 ,...,a k ) k <br />
A n n <br />
D n <br />
[G : H] H G <br />
L H H G <br />
R H H G <br />
a ∤ b a b <br />
d(, ) <br />
d <br />
w() <br />
M m×n ( 2 ) m × n Z 2 <br />
(H) H <br />
δ ij <br />
G ∼ = H G H <br />
(G) G <br />
i g i g (x) =gxg −1 <br />
(G) G <br />
ρ g <br />
G/N G N <br />
G ′ G <br />
φ φ <br />
(a ij ) <br />
O(n) <br />
‖‖ <br />
SO(n) <br />
E(n) <br />
O x x <br />
X g g <br />
G x x <br />
N(H) H <br />
H <br />
Z[i] <br />
R R <br />
Z (p) p <br />
f(x)
R[x] R <br />
R[x 1 ,x 2 ,...,x n ] n <br />
φ α α <br />
Q(x) Q <br />
ν(a) a <br />
F (x) x <br />
F (x 1 ,...,x n ) x 1 ,...,x n <br />
a ≼ b a b <br />
a ∨ b a b <br />
a ∧ b a b <br />
I <br />
O <br />
a ′ a <br />
V V <br />
U ⊕ V U V <br />
(V,W) U V <br />
V ∗ V <br />
F (α 1 ,...,α n ) F α 1 ,...,α n <br />
[E : F ] E F <br />
(p n ) p n <br />
F ∗ F <br />
G(E/F ) E F <br />
F {σi } σ i <br />
F G G <br />
Δ 2
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