Abstract Algebra- Theory and Applications, 2016a

1
S
S S
3+56− 13 + 8/2
2+3=5
2x =6 x =4
ax 2 + bx + c =0 a ≠0
x 3 − 4x 2 +5x − 6
x = −b ± √ b 2 − 4ac
.
2a
2x =6 x =4
2 · 4 6 ≠8

10/5 =
2
p q p q
p
q
ax 2 + bx + c =0 a ≠0
x = −b ± √ b 2 − 4ac
.
2a
ax 2 + bx + c =0 a ≠0
x = −b ± √ b 2 − 4ac
.
2a
ax 2 + bx + c =0 a ≠0
ax 2 + bx + c =0
x 2 + b a x = − c a
x 2 + b ( ) b 2 ( b 2
a x + = −
2a 2a) c a
(
x + b ) 2
= b2 − 4ac
2a 4a 2
x + b
2a = ±√ b 2 − 4ac
2a
x = −b ± √ b 2 − 4ac
.
2a

s r = s
p q q
p
x x
A X a A a ∈ A
x
X = {x 1 ,x 2 ,...,x n }
x 1 ,x 2 ,...,x n
X = {x : x P}
x X P E
E
E = {2, 4, 6,...} E = {x : x x>0}.

2 ∈ E E −3 /∈ E −3
E
N = {n : n } = {1, 2, 3,...};
Z = {n : n } = {...,−1, 0, 1, 2,...};
Q = {r : r } = {p/q : p, q ∈ Z q ≠0};
R = {x : x };
C = {z : z }.
A B A ⊂ B B ⊃ A A B
{4, 5, 8} ⊂{2, 3, 4, 5, 6, 7, 8, 9}
N ⊂ Z ⊂ Q ⊂ R ⊂ C.
B A B ⊂ A
B ≠ A A B A ⊄ B {4, 7, 9} ⊄{2, 4, 5, 8, 9}
A = B A ⊂ B B ⊂ A
∅
A ∪ B A B
A B
A = {1, 3, 5} B = {1, 2, 3, 9}
A ∪ B = {x : x ∈ A x ∈ B};
A ∩ B = {x : x ∈ A x ∈ B}.
A ∪ B = {1, 2, 3, 5, 9} A ∩ B = {1, 3}.
n⋃
A i = A 1 ∪ ...∪ A n
i=1
n⋂
A i = A 1 ∩ ...∩ A n
i=1
A 1 ,...,A n
E O E O
A B A ∩ B = ∅
U
A ⊂ U A A ′
A ′ = {x : x ∈ U x /∈ A}.
A B
A \ B = A ∩ B ′ = {x : x ∈ A x /∈ B}.

R
A = {x ∈ R :0

A ∪ B = ∅ A B
(A ∪ B) ′ ⊂ A ′ ∩ B ′ (A ∪ B) ′ ⊃ A ′ ∩ B ′
x ∈ (A ∪ B) ′ x /∈ A ∪ B x A B
x ∈ A ′ x ∈ B ′
x ∈ A ′ ∩ B ′ (A ∪ B) ′ ⊂ A ′ ∩ B ′
x ∈ A ′ ∩ B ′ x ∈ A ′ x ∈ B ′
x /∈ A x /∈ B x /∈ A ∪ B x ∈ (A ∪ B) ′ (A ∪ B) ′ ⊃ A ′ ∩ B ′
(A ∪ B) ′ = A ′ ∩ B ′
(A \ B) ∩ (B \ A) =∅.
(A \ B) ∩ (B \ A) =(A ∩ B ′ ) ∩ (B ∩ A ′ )
= A ∩ A ′ ∩ B ∩ B ′
= ∅.
A B A × B A
B
A × B = {(a, b) :a ∈ A b ∈ B}.
A = {x, y} B = {1, 2, 3} C = ∅ A × B
{(x, 1), (x, 2), (x, 3), (y, 1), (y, 2), (y, 3)}
A × C = ∅.
n
A 1 ×···×A n = {(a 1 ,...,a n ):a i ∈ A i i =1,...,n}.
A = A 1 = A 2 = ···= A n A n A ×···×A A
n R 3
A×B f ⊂ A×B
A B (a, b) ∈ f
a ∈ A b ∈ B
A f B f : A → B A → f B
(a, b) ∈ A × B f(a) =b f : a ↦→ b
A f
f(A) ={f(a) :a ∈ A} ⊂B
f

A
f
B
a
b
c
A
g
B
a
b
c
A = {1, 2, 3} B = {a, b, c} f
g A B f g 1 ∈ A
B g(1) = a g(1) = b
f : A → B
f : R → R
f(x) =x 3 f : x ↦→ x 3
f : Q → Z f(p/q) =p 1/2 = 2/4
f(1/2) = 1 2
f : A → B f B f(A) =B f
a ∈ A b ∈ B f(a) =b
f a 1 ≠ a 2 f(a 1 ) ≠ f(a 2 )
f(a 1 )=f(a 2 ) a 1 = a 2
f : Z → Q f(n) =n/1 f
g : Q → Z g(p/q) =p p/q
g
f : A → B g : B → C
f g A C (g ◦ f)(x) =g(f(x))

A B C
f
g
a
X
b
c
Y
Z
A
g ◦ f
C
X
Y
Z
f : A → B g : B → C
g ◦ f : A → C
f(x) =x 2 g(x) =2x +5
(f ◦ g)(x) =f(g(x)) = (2x +5) 2 =4x 2 +20x +25
(g ◦ f)(x) =g(f(x)) = 2x 2 +5.
f ◦ g ≠ g ◦ f
f ◦ g = g ◦ f f(x) =x 3 g(x) = 3√ x
(f ◦ g)(x) =f(g(x)) = f( 3√ x )=( 3√ x ) 3 = x
(g ◦ f)(x) =g(f(x)) = g(x 3 )= 3√ x 3 = x.
2 × 2
( ) a b
A = ,
c d
T A : R 2 → R 2
T A (x, y) =(ax + by, cx + dy)
(x, y) R 2
( ) a b x
=
c d)(
y
( ) ax + by
.
cx + dy
R n R m

S = {1, 2, 3} π : S → S
π(1) = 2, π(2) = 1, π(3) = 3.
π
( )
1 2 3
=
π(1) π(2) π(3)
( ) 1 2 3
.
2 1 3
S π : S → S S
f : A → B g : B → C h : C → D
(h ◦ g) ◦ f = h ◦ (g ◦ f)
f g g ◦ f
f g g ◦ f
f g g ◦ f
h ◦ (g ◦ f) =(h ◦ g) ◦ f.
a ∈ A
(h ◦ (g ◦ f))(a) =h((g ◦ f)(a))
= h(g(f(a)))
=(h ◦ g)(f(a))
=((h ◦ g) ◦ f)(a).
f g c ∈ C
a ∈ A (g ◦ f)(a) =g(f(a)) = c g
b ∈ B g(b) =c a ∈ A f(a) =b
(g ◦ f)(a) =g(f(a)) = g(b) =c.
S id S id S
id(s) =s s ∈ S g : B → A
f : A → B g ◦ f = id A f ◦ g = id B
f −1 f
f(x) =x 3 f −1 (x) = 3√ x
f(x) = x
f −1 (x) =e x
f(f −1 (x)) = f(e x )= e x = x
f −1 (f(x)) = f −1 ( x) =e x = x

A R 2 R 2
A =
( ) 3 1
.
5 2
T A (x, y) =(3x + y, 5x +2y).
T A A T −1
A
= T A −1
( )
A −1 2 −1
=
;
−5 3
T −1
A
T −1
A
(x, y) =(2x − y, −5x +3y).
◦ T A(x, y) =T A ◦ T −1 (x, y) =(x, y).
A
T B (x, y) =(3x, 0)
( ) 3 0
B = ,
0 0
T −1
B
(x, y) =(ax + by, cx + dy)
(x, y) =T ◦ T −1
B
(x, y) =(3ax +3by, 0)
x y y 0
π =
( 1 2
) 3
2 3 1
S = {1, 2, 3}
( )
π −1 1 2 3
=
3 1 2
π
f : A → B g : B → A
g ◦ f = id A g(f(a)) = a a 1 ,a 2 ∈ A f(a 1 )=f(a 2 )
a 1 = g(f(a 1 )) = g(f(a 2 )) = a 2 f b ∈ B
f a ∈ A f(a) =b f(g(b)) = b
g(b) ∈ A a = g(b)
f b ∈ B f a ∈ A
f(a) =b f a g g(b) =a
f

X
R ⊂ X × X
(x, x) ∈ R x ∈ X
(x, y) ∈ R (y, x) ∈ R
(x, y) (y, z) ∈ R (x, z) ∈ R
R X x ∼ y (x, y) ∈ R
= ≡ ∼ =
p q r s q s p/q ∼ r/s
ps = qr ∼
p/q ∼ r/s r/s ∼ t/u q s u ps = qr ru = st
psu = qru = qst.
s ≠0 pu = qt p/q ∼ t/u
f g R
f(x) ∼ g(x) f ′ (x) =g ′ (x)
∼ f(x) ∼ g(x)
g(x) ∼ h(x) f(x) − g(x) =c 1 g(x) − h(x) =c 2
c 1 c 2
f(x) − h(x) =(f(x) − g(x)) + (g(x) − h(x)) = c 1 − c 2
f ′ (x) − h ′ (x) =0 f(x) ∼ h(x)
(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
∼ R 2
A B 2 × 2
2 × 2 A ∼ B
P PAP −1 = B
A =
( 1
) 2
−1 1
( ) −18 33
B =
,
−11 20
A ∼ B PAP −1 = B
P =
( ) 2 5
.
1 3
I 2 × 2
I =
( ) 1 0
.
0 1
IAI −1 = IAI = A
A ∼ B P PAP −1 = B
A = P −1 BP = P −1 B(P −1 ) −1 .

A ∼ B B ∼ C P Q
PAP −1 = B QBQ −1 = C
C = QBQ −1 = QP AP −1 Q −1 =(QP )A(QP ) −1 ,
P X X 1 ,X 2 ,... X i ∩X j =
∅ i ≠ j ⋃ k X k = X ∼ X x ∈ X
[x] ={y ∈ X : y ∼ x} x
∼ X X
X P = {X i } X
X X i
∼ X x ∈ X
x ∈ [x] [x] X = ⋃ x∈X [x]
x, y ∈ X [x] =[y] [x] ∩ [y] =∅
[x] [y] z ∈ [x] ∩ [y] z ∼ x z ∼ y
x ∼ y [x] ⊂ [y] [y] ⊂ [x] [x] =[y]
P = {X i } X
x
y y x x ∼ y y ∼ x
x y y z x
z
(p, q)
(r, s)
f(x) g(x)
R 2 (x 1 ,y 1 ) ∼ (x 2 ,y 2 ) x 2 1 + y2 1 =
x 2 2 + y2 2
r s n ∈ N r
s n r s n r − s n
r − s = nk k ∈ Z r ≡ s ( n)
41 ≡ 17 ( 8) 41 − 17 = 24 8
n Z r
r − r =0 n r ≡ s

( n) r − s = −(s − r) n s − r n s ≡ r
( n) r ≡ s ( n) s ≡ t ( n)
k l r − s = kn s − t = ln
r − t n
r − t = r − s + s − t = kn + ln =(k + l)n,
r − t n
3
[0] = {...,−3, 0, 3, 6,...},
[1] = {...,−2, 1, 4, 7,...},
[2] = {...,−1, 2, 5, 8,...}.
[0] ∪ [1] ∪ [2] = Z [0] [1] [2]
n
n
A = {x : x ∈ N x },
B = {x : x ∈ N x },
C = {x : x ∈ N x 5}.
A ∩ B
B ∩ C
A ∪ B
A ∩ (B ∪ C)
A = {a, b, c} B = {1, 2, 3} C = {x} D = ∅
A × B
B × A
A × B × C
A × D
A B A × B = B × A
A ∪∅= A A ∩∅= ∅
A ∪ B = B ∪ A A ∩ B = B ∩ A
A ∪ (B ∩ C) =(A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) =(A ∩ B) ∪ (A ∩ C)
A ⊂ B A ∩ B = A
(A ∩ B) ′ = A ′ ∪ B ′
A ∪ B =(A ∩ B) ∪ (A \ B) ∪ (B \ A)

(A ∪ B) × C =(A × C) ∪ (B × C)
(A ∩ B) \ B = ∅
(A ∪ B) \ B = A \ B
A \ (B ∪ C) =(A \ B) ∩ (A \ C)
A ∩ (B \ C) =(A ∩ B) \ (A ∩ C)
(A \ B) ∪ (B \ A) =(A ∪ B) \ (A ∩ B)
f : Q → Q
f
f(p/q) = p +1
p − 2
f(p/q) = p + q
q 2
f(p/q) = 3p
3q
f(p/q) = 3p2
7q 2 − p q
f : R → R f(x) =e x
f : Z → Z f(n) =n 2 +3
f : R → R f(x) = x
f : Z → Z f(x) =x 2
f : A → B g : B → C
f −1 g −1 (g ◦ f) −1 = f −1 ◦ g −1
f : N → N
f : N → N
R 2 (x 1 ,y 1 ) ∼ (x 2 ,y 2 ) x 2 1 + y2 1 = x2 2 + y2 2
f : A → B g : B → C
f g g ◦ f
g ◦ f g
g ◦ f f
g ◦ f f g
g ◦ f g f
f(x) = x +1
x − 1 .
f f f ◦ f −1
f −1 ◦ f
f : X → Y A 1 ,A 2 ⊂ X B 1 ,B 2 ⊂ Y

f(A 1 ∪ A 2 )=f(A 1 ) ∪ f(A 2 )
f(A 1 ∩ A 2 ) ⊂ f(A 1 ) ∩ f(A 2 )
f −1 (B 1 ∪ B 2 )=f −1 (B 1 ) ∪ f −1 (B 2 )
f −1 (B) ={x ∈ X : f(x) ∈ B}.
f −1 (B 1 ∩ B 2 )=f −1 (B 1 ) ∩ f −1 (B 2 )
f −1 (Y \ B 1 )=X \ f −1 (B 1 )
x ∼ y R x ≥ y
m ∼ n Z mn > 0
x ∼ y R |x − y| ≤4
m ∼ n Z m ≡ n ( 6)
∼ R 2 (a, b) ∼ (c, d) a 2 +b 2 ≤ c 2 +d 2
∼
m × n R n R m
x ∼ y
y ∼ x x ∼ x
R 2 \{(0, 0)} (x 1 ,y 1 ) ∼ (x 2 ,y 2 )
λ (x 1 ,y 1 )=(λx 2 ,λy 2 ) ∼
R 2 \(0, 0)
P(R)

2
1+2+···+ n =
n(n +1)
2
n n =1
2 3 4
n
(n +1)
n =1
1(1 + 1)
1= .
2
n
n(n +1)
1+2+···+ n +(n +1)= + n +1
2
= n2 +3n +2
2
(n +1)[(n +1)+1]
= .
2
(n +1)
S N
S
S(n)
n ∈ N S(n 0 ) n 0 k
k ≥ n 0 S(k) S(k +1) S(n) n
n 0

n ≥ 3 2 n >n+4
8=2 3 > 3+4=7,
n 0 =3 2 k >k+4 k ≥ 3 2 k+1 =2· 2 k >
2(k +4)
2(k +4)=2k +8>k+5=(k +1)+4
k n ≥ 3
10 n+1 +3· 10 n +5 9 n ∈ N n =1
10 1+1 +3· 10 + 5 = 135 = 9 · 15
9 10 k+1 +3· 10 k +5 9 k ≥ 1
9
10 (k+1)+1 +3· 10 k+1 +5=10 k+2 +3· 10 k+1 +50− 45
= 10(10 k+1 +3· 10 k +5)− 45
(a + b) n =
a b n ∈ N
( n
k)
=
n∑
k=0
( n
k)
a k b n−k ,
n!
k!(n − k)!
( ) n +1
=
k
( n
k)
+
( ) n
.
k − 1
( ( )
n n
+ =
k)
k − 1
n!
k!(n − k)! + n!
(k − 1)!(n − k +1)!
(n +1)!
=
k!(n +1− k)!
( ) n +1
= .
k
n =1 n
1
(a + b) n+1 =(a + b)(a + b) n
( n∑ ( )
n
=(a + b)
k)a k b n−k
k=0
n∑
( n
n∑
( n
= a
k)
k+1 b n−k + a
k)
k b n+1−k
k=0
k=0

= a n+1 +
= a n+1 +
k=0
n∑
( ) n
a k b n+1−k +
k − 1
n∑
[( n
)
k − 1
k=1
k=1
n+1
∑
( ) n +1
=
a k b n+1−k .
k
n∑
k=1
( n
k)
a k b n+1−k + b n+1
( n
+ a
k)]
k b n+1−k + b n+1
S(n)
n ∈ N S(n 0 ) n 0 S(n 0 ),S(n 0 +
1),...,S(k) S(k +1) k ≥ n 0 S(n)
n ≥ n 0
S Z S
Z
1
S = {n ∈ N : n ≥ 1} 1 ∈ S n ∈ S 0 < 1
n = n +0

a b b>0
q r
a = bq + r
0 ≤ r0 a − b · 0 ∈ S
a0.
a − b(q +1) S a − b(q +1)

= a − (ar + bs)q
= a − arq − bsq
= a(1 − rq)+b(−sq),
S d S
r ′ =0 d a d b d
a b
d ′ a b d ′ | d
a = d ′ h b = d ′ k
d = ar + bs = d ′ hr + d ′ ks = d ′ (hr + ks).
d ′ d d a b
a b
r s ar + bs =1
945 2415
2415 = 945 · 2 + 525
945 = 525 · 1 + 420
525 = 420 · 1 + 105
420 = 105 · 4+0.
105 420 105 525 105 945 105 2415
105 945 2415 d 945 2415
d 105 (945, 2415) = 105
r s 945r + 2415s = 105
105 = 525 + (−1) · 420
= 525 + (−1) · [945 + (−1) · 525]
=2· 525 + (−1) · 945
=2· [2415 + (−2) · 945] + (−1) · 945
=2· 2415 + (−5) · 945.
r = −5 s =2 r s r =41 s = −16
(a, b) =d
r 1 >r 2 > ···>r n = d
b = aq 1 + r 1
a = r 1 q 2 + r 2
r 1 = r 2 q 3 + r 3

n−2 = r n−1 q n + r n
r n−1 = r n q n+1 .
r s ar + bs = d
d = r n
= r n−2 − r n−1 q n
= r n−2 − q n (r n−3 − q n−1 r n−2 )
= −q n r n−3 +(1+q n q n−1 )r n−2
= ra + sb.
d
a b d a b
p p>1 p p
p 1 p n>1
a b p p | ab
p | a p | b
p a p | b (a, p) =1
r s ar + ps =1
b = b(ar + ps) =(ab)r + p(bs).
p ab p b =(ab)r + p(bs)
p 1 ,p 2 ,...,p n P = p 1 p 2 ···p n +1 P
p i 1 ≤ i ≤ n p i P − p 1 p 2 ···p n =1
P p ≠ p i
P
n
n>1
n = p 1 p 2 ···p k ,
p 1 ,...,p k
n = q 1 q 2 ···q l ,
k = l q i p i

n
n =2 n
m 1 ≤ m

1 2 +2 2 + ···+ n 2 n(n + 1)(2n +1)
=
6
n ∈ N
1 3 +2 3 + ···+ n 3 = n2 (n +1) 2
4
n ∈ N
n! > 2 n n ≥ 4
n(3n − 1)x
x +4x +7x + ···+(3n − 2)x =
2
n ∈ N
10 n+1 +10 n +1 3 n ∈ N
4 · 10 2n +9· 10 2n−1 +5 99 n ∈ N
n√
a1 a 2 ···a n ≤ 1 n∑
a k .
n
k=1
f (n) (x) f (n) n f
(fg) (n) (x) =
n∑
k=0
( n
k)
f (k) (x)g (n−k) (x).
1+2+2 2 + ···+2 n =2 n+1 − 1 n ∈ N
1
2 + 1 6 + ···+ 1
n(n +1) =
n
n +1
n ∈ N
x (1+x) n −1 ≥ nx n =0, 1, 2,...
X X P(X)
X
P({a, b}) ={∅, {a}, {b}, {a, b}}.
n n
2 n
S ⊂ N
1 ∈ S n +1∈ S n ∈ S S = N
a b (a, b)
r s (a, b) =ra + sb

14 39
234 165
1739 9923
471 562
23771 19945
−4357 3754
a b r s ar + bs =1
a b
1, 1, 2, 3, 5, 8, 13, 21,....
f 1 =1 f 2 =1 f n+2 = f n+1 + f n n ∈ N
f n < 2 n
f n+1 f n−1 = f 2 n +(−1) n n ≥ 2
f n =[(1+ √ 5) n − (1 − √ 5) n ]/2 n√ 5
n→∞ f n /f n+1 =( √ 5 − 1)/2
f n f n+1
a b (a, b) =1 r s
ar + bs =1
(a, s) =(r, b) =(r, s) =1.
x, y ∈ N xy x y
4k
4k +1 k
a, b, r, s
a 2 + b 2 = r 2
a 2 − b 2 = s 2 .
a r s b
n ∈ N
n 0, 1,...,n− 1 r
s Z 0 ≤ s

p ≥ 2 2 p − 1 p
6n +5
4n − 1
2 p q
p 2 =2q 2 √ 2
N n 1

3
Z
2 × 2
2 × 2
n
n
a b n n
a − b n Z n
Z n 12
[0] = {...,−12, 0, 12, 24,...},
[1] = {...,−11, 1, 13, 25,...},
[11] = {...,−1, 11, 23, 35,...}.
0, 1,...,11
[0], [1],...,[11] Z n a b
n (a + b) ( n) a + b
n n (ab) ( n) ab
n

n
7+4≡ 1 ( 5) 7 · 3 ≡ 1 ( 5)
3+5≡ 0 ( 8) 3 · 5 ≡ 7 ( 8)
3+4≡ 7 ( 12) 3 · 4 ≡ 0 ( 12)
n
0 n
Z n
Z 8 2 4 6
n =2 4 6 k
kn ≡ 1( 8)
· 0 1 2 3 4 5 6 7
0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7
2 0 2 4 6 0 2 4 6
3 0 3 6 1 4 7 2 5
4 0 4 0 4 0 4 0 4
5 0 5 2 7 4 1 6 3
6 0 6 4 2 0 6 4 2
7 0 7 6 5 4 3 2 1
Z 8
Z n
a, b, c ∈ Z n
n
a + b ≡ b + a ( n)
ab ≡ ba ( n).
(a + b)+c ≡ a +(b + c) ( n)
(ab)c ≡ a(bc) ( n).
a +0≡ a ( n)
a · 1 ≡ a ( n).
a(b + c) ≡ ab + ac
( n).

a −a
a +(−a) ≡ 0
( n).
a (a, n) =1
b a ( n) b
ab ≡ 1
( n).
n a + b
n b + a n
(a, n) =1 r s ar + ns =1
ns =1− ar ar ≡ 1( n) b
r ab ≡ 1( n)
b ab ≡ 1( n) n
ab − 1 k ab − nk =1 d = (a, n) d
ab − nk d 1 d =1
A
B
A
B
D
C
D
C
A
D
B
C
180 ◦
C
B
D
A
A
D
B
B
C
C
A
D
A
D
B
D
C
A
C
B
180 ◦ 360 ◦

90 ◦
A
B
C A
B
C
id =
( A B
) C
A B C
A
B
C C
A
B
ρ 1 =
( A B
) C
B C A
A
B
C B
C
A
ρ 2 =
( A B
) C
C A B
A
B
C A
C
B
μ 1 =
( A B
) C
A C B
A
B
C C
B
A
μ 2 =
( A B
) C
C B A
A
B
C B
A
C
μ 3 =
( A B
) C
B A C
△ABC
△ABC A B C
S π : S → S 3! = 6
3·2·1 =3!=6
A B B C C A
( ) A B C
.
B C A
120 ◦
△ABC
μ 1 ρ 1
ρ 1 μ 1

(μ 1 ρ 1 )(A) =μ 1 (ρ 1 (A)) = μ 1 (B) =C
(μ 1 ρ 1 )(B) =μ 1 (ρ 1 (B)) = μ 1 (C) =B
(μ 1 ρ 1 )(C) =μ 1 (ρ 1 (C)) = μ 1 (A) =A.
μ 2
ρ 1 μ 1 μ 3
ρ 1 μ 1 ≠ μ 1 ρ 1 △ABC
α β αβ =
◦ ρ 1 ρ 2 μ 1 μ 2 μ 3
ρ 1 ρ 2 μ 1 μ 2 μ 3
ρ 1 ρ 1 ρ 2 μ 3 μ 1 μ 2
ρ 2 ρ 2 ρ 1 μ 2 μ 3 μ 1
μ 1 μ 1 μ 2 μ 3 ρ 1 ρ 2
μ 2 μ 2 μ 3 μ 1 ρ 2 ρ 1
μ 3 μ 3 μ 1 μ 2 ρ 1 ρ 2
n
G G × G → G
(a, b) ∈ G × G a ◦ b ab G
a b (G, ◦) G (a, b) ↦→ a ◦ b
a, b, c ∈ G
(a ◦ b) ◦ c = a ◦ (b ◦ c)
e ∈ G
a ∈ G
e ◦ a = a ◦ e = a.
a ∈ G a −1
a ◦ a −1 = a −1 ◦ a = e.
G a ◦ b = b ◦ a a, b ∈ G

Z = {...,−1, 0, 1, 2,...}
m, n ∈ Z
+
◦ m+n m◦n 0
n ∈ Z −n n −1
m + n = n + m
ab a ◦ b
m + n −n
m − n m +(−n)
n n
Z 5 0 1 2 3 4
Z 5
m + n
Z 5 2+3=3+2=0
Z 5 Z n = {0, 1,...,n− 1}
n
+ 0 1 2 3 4
0 0 1 2 3 4
1 1 2 3 4 0
2 2 3 4 0 1
3 3 4 0 1 2
4 4 0 1 2 3
(Z 5 , +)
Z n Z n
1 · k = k · 1=k k ∈ Z n
0 0 · k = k · 0=0 k Z n
Z n \{0} 2 ∈ Z 6
0 · 2=0 1· 2=2
2 · 2=4 3· 2=0
4 · 2=2 5· 2=4.
k Z n k
n Z n U(n) U(n)
Z n U(8)

· 1 3 5 7
1 1 3 5 7
3 3 1 7 5
5 5 7 1 3
7 7 5 3 1
U(8)
αβ = βα
α β
S 3 D 3
M 2 (R) 2 × 2 GL 2 (R)
M 2 (R)
( ) a b
A =
c d
GL 2 (R) A −1 AA −1 = A −1 A = I I 2 × 2
A
A A = ad − bc ≠0
( ) 1 0
I = .
0 1
A ∈ GL 2 (R)
A −1 =
1
ad − bc
( d −b
−c a
AB = BA
GL 2 (R)
( ) ( )
1 0
0 1
1=
I =
J =
0 1
( ) 0 i
i 0
K =
)
.
−1 0
( ) i 0
,
0 −i
i 2 = −1 I 2 = J 2 = K 2 = −1 IJ = K JK = I KI = J
JI = −K KJ = −I IK = −J Q 8 = {±1, ±I,±J, ±K}
Q 8
C ∗
C ∗ 1 z = a + bi
z −1 = a − bi
a 2 + b 2
z

G n
|G| = n Z 5 5 Z
|Z| = ∞
G
e ∈ G eg = ge = g g ∈ G
e e ′ G eg = ge = g e ′ g = ge ′ = g
g ∈ G e = e ′ e ee ′ = e ′
e ′ ee ′ = e e = ee ′ = e ′
g ′ g ′′ g
G gg ′ = g ′ g = e gg ′′ = g ′′ g = e g ′ = g ′′
g ′ = g ′ e = g ′ (gg ′′ )=(g ′ g)g ′′ = eg ′′ = g ′′
g G g g −1
G a, b ∈ G (ab) −1 = b −1 a −1
a, b ∈ G abb −1 a −1 = aea −1 = aa −1 = e b −1 a −1 ab = e
(ab) −1 = b −1 a −1
G a ∈ G (a −1 ) −1 = a
a −1 (a −1 ) −1 = e
a
(a −1 ) −1 = e(a −1 ) −1 = aa −1 (a −1 ) −1 = ae = a.
a b
G x ∈ G ax = b
x
G a b G
ax = b xa = b G
ax = b x
ax = b a −1 x = ex = a −1 ax = a −1 b
x 1 x 2 ax = b ax 1 =
b = ax 2 x 1 = a −1 ax 1 = a −1 ax 2 = x 2
xa = b
G a, b, c ∈ G ba = ca b = c ab = ac
b = c

G
g ∈ G g 0 = e n ∈ N
g n = g · g ···g
} {{ }
n
g −n = g −1 · g −1 ···g −1 .
} {{ }
n
g, h ∈ G
g m g n = g m+n m, n ∈ Z
(g m ) n = g mn m, n ∈ Z
(gh) n =(h −1 g −1 ) −n n ∈ Z G (gh) n = g n h n
(gh) n ≠ g n h n
Z Z n
ng
g n
mg + ng =(m + n)g m, n ∈ Z
m(ng) =(mn)g m, n ∈ Z
m(g + h) =mg + mh n ∈ Z
Z Z n

2Z = {...,−2, 0, 2, 4,...}
H G H G
G H H G
H = {e} G
G
R ∗
1 a ∈ R ∗
1/a
Q ∗ = {p/q : p q }
R ∗ R ∗ 1 1 = 1/1
R ∗ Q ∗ Q ∗ p/q r/s
pr/qs Q ∗ p/q ∈ Q ∗ Q ∗
(p/q) −1 = q/p R ∗ Q ∗
C ∗
H = {1, −1,i,−i} H C ∗ H
H ⊂ C ∗
SL 2 (R) GL 2 (R)
( ) a b
A =
c d
SL 2 (R) ad − bc =1 SL 2 (R)
2×2
SL 2 (R) A
( )
A −1 d −b
=
.
−c a
SL 2 (R)
H G
G H G G
2 × 2 M 2 (R)
2 × 2 M 2 (R)
M 2 (R)
( 1 0
0 1
)
+
GL 2 (R)
( −1 0
0 −1
)
=
( 0 0
0 0
)
,

Z 4
0 2 Z 2
Z 2 × Z 2
(a, b) +(c, d) =(a + c, b + d)
Z 2 × Z 2 Z 2 × Z 2
H 1 = {(0, 0), (0, 1)} H 2 = {(0, 0), (1, 0)} H 3 = {(0, 0), (1, 1)} Z 4 Z 2 × Z 2
+ (0, 0) (0, 1) (1, 0) (1, 1)
(0, 0) (0, 0) (0, 1) (1, 0) (1, 1)
(0, 1) (0, 1) (0, 0) (1, 1) (1, 0)
(1, 0) (1, 0) (1, 1) (0, 0) (0, 1)
(1, 1) (1, 1) (1, 0) (0, 1) (0, 0)
Z 2 × Z 2
H G
e G H
h 1 ,h 2 ∈ H h 1 h 2 ∈ H
h ∈ H h −1 ∈ H
H G
H e H e H = e
e G e H e H = e H ee H = e H e = e H
ee H = e H e H e = e H
H h ∈ H H
h ′ ∈ H hh ′ = h ′ h = e G
h ′ = h −1
H
G
H G H G
H ≠ ∅ g, h ∈ H gh −1 H
H G gh −1 ∈ H
g h H h H h −1 H
gh −1 ∈ H
H ⊂ G H ≠ ∅ gh −1 ∈ H g, h ∈ H
g ∈ H gg −1 = e H g ∈ H eg −1 = g −1 H h 1 ,h 2 ∈ H
H h 1 (h −1
2 )−1 = h 1 h 2 ∈ H
H G

x ∈ Z
3x ≡ 2( 7)
5x +1≡ 13 ( 23)
5x +1≡ 13 ( 26)
9x ≡ 3( 5)
5x ≡ 1( 6)
3x ≡ 1( 6)
G = {a, b, c, d}
◦ a b c d
a a c d a
b b b c d
c c d a b
d d a b c
◦ a b c d
a a b c d
b b c d a
c c d a b
d d a b c
◦ a b c d
a a b c d
b b a d c
c c d a b
d d c b a
◦ a b c d
a a b c d
b b a c d
c c b a d
d d d b c
(Z 4 , +)
D 4
U(12)
S = R \{−1} S a ∗ b = a + b + ab
(S, ∗)
A B GL 2 (R) AB ≠ BA
SL 2 (R)

⎛
1 x
⎞
y
⎝0 1 z⎠
0 0 1
⎛ ⎞ ⎛
1 x y 1 x ′ y ′ ⎞ ⎛
1 x + x ′ y + y ′ + xz ′ ⎞
⎝
0 1 z
0 0 1
⎠ ⎝
0 1 z ′
0 0 1
⎠ = ⎝
0 1 z + z ′
0 0 1
(AB) =(A) (B) GL 2 (R)
GL 2 (R) A B GL 2 (R)
AB ∈ GL 2 (R)
Z n 2 = {(a 1,a 2 ,...,a n ):a i ∈ Z 2 } Z n 2
(a 1 ,a 2 ,...,a n )+(b 1 ,b 2 ,...,b n )=(a 1 + b 1 ,a 2 + b 2 ,...,a n + b n ).
Z n 2
R ∗ = R \{0}
R ∗ Z G = R ∗ × Z ◦ G
(a, m) ◦ (b, n) =(ab, m + n) G
G g, h ∈ G (gh) n ≠ g n h n
n! n
0+a ≡ a +0≡ a ( n)
a ∈ Z n
n
⎠ .
a · 1 ≡ a
( n).
a ∈ Z n b ∈ Z n
a + b ≡ b + a ≡ 0
( n).
n
n
n
n
a(b + c) ≡ ab + ac
( n).
a b G ab n a −1 =(aba −1 ) n n ∈ Z

U(n) Z n n>2
k ∈ U(n) k 2 =1 k ≠1
g 1 g 2 ···g n gn
−1 gn−1 −1 1
G a, b ∈ G
xa = b G
G
G ba = ca b = c ab = ac b = c a, b, c ∈ G
a 2 = e a G G
G a ∈ G a
a 2 = e
G (ab) 2 = a 2 b 2 a b G G
Z 3 × Z 3 Z 3 × Z 3
Z 9
H = {2 k : k ∈ Z} H Q ∗
n =0, 1, 2,... nZ = {nk : k ∈ Z} nZ Z
Z
T = {z ∈ C ∗ : |z| =1} T C ∗
G 2 × 2
( )
θ − θ
,
θ θ
θ ∈ R G SL 2 (R)
G = {a + b √ 2:a, b ∈ Q a b }
R ∗
G 2 × 2
{( ) }
a b
H = : a + d =0 .
c d
H G
SL 2 (Z) 2 × 2
SL 2 (R)
Q 8
G G
H K G H ∪K
G
H K G HK = {hk : h ∈
H k ∈ K} G G

G g ∈ G
Z(G) ={x ∈ G : gx = xg g ∈ G}
G G
a b G a 4 b = ba a 3 = e ab = ba
xy = x −1 y −1 x y G G
H G
C(H) ={g ∈ G : gh = hg h ∈ H}.
C(H) G H G
H G g ∈ G gHg −1 = {ghg −1 : h ∈ H}
G
d 1 d 2 ···d 12
3 · d 1 +1· d 2 +3· d 3 + ···+3· d 11 +1· d 12 ≡ 0 ( 10).
d 12

(d 1 ,d 2 ,...,d k ) · (w 1 ,w 2 ,...,w k ) ≡ 0 ( n)
d 1 w 1 + d 2 w 2 + ···+ d k w k ≡ 0
( n).
(d 1 ,d 2 ,...,d k ) · (w 1 ,w 2 ,...,w k ) ≡ 0( n)
k d 1 d 2 ···d k 0 ≤ d i

4
Z Z n
3 ∈ Z
3
3Z = {...,−3, 0, 3, 6,...}.
3Z
3
3 3
H = {2 n : n ∈ Z} H
Q ∗ a =2 m b =2 n H ab −1 =2 m 2 −n =2 m−n
H H Q ∗ 2
G a G
〈a〉 = {a k : k ∈ Z}
G 〈a〉 G a
〈a〉 a 0 = e g h 〈a〉
〈a〉 g = a m h = a n m n
gh = a m a n = a m+n 〈a〉 g = a n 〈a〉 g −1 = a −n
〈a〉 H G a a
H 〈a〉 〈a〉 G
a
〈a〉 = {na : n ∈ Z}
a ∈ G 〈a〉 a G
a G = 〈a〉 G a G a
G a n
a n = e |a| = n n
a |a| = ∞ a

1 5 Z 6 Z 6
2 ∈ Z 6 3
2 〈2〉 = {0, 2, 4}
Z Z n 1 −1 Z
Z n Z n
Z 6
U(9) Z 9 U(9)
{1, 2, 4, 5, 7, 8} U(9)
2 1 =2 2 2 =4
2 3 =8 2 4 =7
2 5 =5 2 6 =1.
S 3
S 3
S 3
{,ρ 1 ,ρ 2 } {,μ 1 } {,μ 2 } {,μ 3 }
{}
S 3
G a ∈ G G g h G
a g = a r h = a s
G
gh = a r a s = a r+s = a s+r = a s a r = hg,
G G G
G
G a H
G H = {e} H H
g g a n n H

g −1 = a −n H n −n
H a n>0 m
a m ∈ H m
h = a m H h ′ ∈ H
h h ′ ∈ H H G h ′ = a k
k q r k = mq + r
0 ≤ r

C = {a + bi : a, b ∈ R},
i 2 = −1 z = a + bi a z b
z
z = a + bi w = c + di
z + w =(a + bi)+(c + di) =(a + c)+(b + d)i.
i 2 = −1
z w
(a + bi)(c + di) =ac + bdi 2 + adi + bci =(ac − bd)+(ad + bc)i.
z = a + bi
z −1 ∈ C ∗ zz −1 = z −1 z =1z = a + bi
z −1 = a − bi
a 2 + b 2 .
z = a + bi z = a − bi
z = a + bi |z| = √ a 2 + b 2
z =2+3i w =1− 2i
z + w =(2+3i)+(1− 2i) =3+i
zw =(2+3i)(1 − 2i) =8− i.
z −1 = 2
13 − 3 13 i
|z| = √ 13
z =2− 3i.
y
z 3 = −3+2i
z 1 =2+3i
0
x
z 2 =1− 2i

z = a + bi xy a x
b y
z 1 =2+3i z 2 =1− 2i
z 3 = −3+2i
y
r
a + bi
0
θ
x
θ x
r
z = a + bi = r( θ + i θ).
r = |z| = √ a 2 + b 2
a = r θ
b = r θ.
r( θ + i θ) r θ z
0 ◦ ≤ θ

z = r θ w = s φ
zw = rs (θ + φ).
z =3(π/3) w =2(π/6) zw =6(π/2) = 6i
z = r θ
n =1, 2,...
[r θ] n = r n (nθ)
n n =1
k 1 ≤ k ≤ n
z n+1 = z n z
= r n ( nθ + i nθ)r( θ + i θ)
= r n+1 [( nθ θ − nθ θ)+i( nθ θ + nθ θ)]
= r n+1 [(nθ + θ)+i (nθ + θ)]
= r n+1 [(n +1)θ + i (n +1)θ].
z =1+i z 10
(1 + i) 10 z 10
z 10 =(1+i) 10
(√ ( π
)) 10
= 2
4
=( √ ( ) 5π
2) 10
2
( π
)
=32
2
=32i.
C ∗
Q ∗ R ∗ C ∗
T = {z ∈ C : |z| =1}.
C ∗
H = {1, −1,i,−i} H 1 −1 i
−i z 4 =1
z n =1 n

z n =1 n
( ) 2kπ
z = ,
n
k =0, 1,...,n− 1 n T
n
z n =
(
n 2kπ )
= (2kπ) =1.
n
z 2kπ/n
2π z n =1
n n
n T
n n
√ √
2 2
ω =
2 + 2 i
√ √
2 2
ω 3 = −
2 + 2 i
√ √
2 2
ω 5 = −
2 − 2 i
√ √
2 2
ω 7 =
2 − 2 i.
ω 3
i
y
ω
−1
0
x
ω 5
−i
ω 7
2 2
2 8
2 21,000,000 .

2 37,398,332 ( 46,389),
0 46,388
n
a
2
a =2 k 1
+2 k 2
+ ···+2 kn ,
k 1

271 28 ≡ 16 ( 481).
271 321 ≡ 271 20 +2 6 +2 8 ( 481)
≡ 271 20 · 271 26 · 271 28 ( 481)
≡ 271 · 419 · 16 ( 481)
≡ 1,816,784 ( 481)
≡ 47 ( 481).
n
Z 60
U(8)
Q
G G
5 ∈ Z 12
√ 3 ∈ R
√ 3 ∈ R ∗
−i ∈ C ∗
72 ∈ Z 240
312 ∈ Z 471
Z 7
Z 24 15
Z 12
Z 60
Z 13
Z 48
U(20)

U(18)
R ∗ 7
C ∗ i i 2 = −1
C ∗ 2i
C ∗ (1 + i)/ √ 2
C ∗ (1 + √ 3 i)/2
GL 2 (R)
( 0
) 1
−1 0
( ) 0 1/3
3 0
( ) 1 −1
1 0
( ) 1 −1
0 1
( 1
) −1
−1 0
(√ )
3/2 1/2
−1/2 √ 3/2
Z 18
D 4
Q 8
U(30)
Z 32
∗
Z Q ∗ R ∗
a 24 = e G a
n
n ≤ 20 U(n)
( )
( )
0 1
0 −1
A =
B =
−1 0
1 −1
GL 2 (R) A B AB
(3 − 2i)+(5i − 6)
(4 − 5i) − (4i − 4)
(5 − 4i)(7 + 2i)
(9 − i)(9 − i)
i 45
(1 + i)+(1 + i)
a + bi

2 (π/6)
5 (9π/4)
3 (π)
(7π/4)/2
1 − i
−5
2+2i
√ 3+i
−3i
2i +2 √ 3
((1 − i)/2) 4
(1 + i) −1
(−i) 10 (−2+2i) −5
(1 − i) 6
( √ 3+i) 5
(− √ 2 − √ 2 i) 12
|z| = |z|
zz = |z| 2
z −1 = z/|z| 2
|z + w| ≤|z| + |w|
|z − w| ≥||z|−|w||
|zw| = |z||w|
292 3171 ( 582)
2557 341 ( 5681)
2071 9521 ( 4724)
971 321 ( 765)
a, b ∈ G
a a −1
g ∈ G |a| = |g −1 ag|
ab ba
p q Z pq
p r Z p r
Z p p
g h 15 16 G
〈g〉∩〈h〉
a G 〈a m 〉∩〈a n 〉
Z n n>2

G a b ∈ G |a| = m |b| = n
(m, n) =1 〈a〉∩〈b〉 = {e}
G G
G
G n x y = x k
(k, n) =1 y G
G 2
G 4
G pq (p, q) =1G a
b p q G
Z nZ n =0, 1, 2,...
Z n r 1 ≤ r

5
△ABC
S = {A, B, C} π : S → S
( ) ( ) ( )
A B C A B C A B C
A B C
( A B C
A C B
C A B B C A
) ( ) A B C
B A C
) ( A B C
C B A
( A B
) C
B C A
A B B C C A
A ↦→ B
B ↦→ C
C ↦→ A.
X S X X
X = {1, 2,...,n} S n S X
S n n
n S n n!
S n 1 1 2 2 ... n n
f : S n → S n f −1 f
|S n | = n!

S n
G S 5
( )
1 2 3 4 5
σ =
1 2 3 5 4
( )
1 2 3 4 5
τ =
3 2 1 4 5
( )
1 2 3 4 5
μ =
.
3 2 1 5 4
G
◦ σ τ μ
σ τ μ
σ σ μ τ
τ τ μ σ
μ μ τ σ
σ τ X
σ τ (σ ◦τ)(x) =σ(τ(x)) τ
σ
στ τ σ
στ(x) σ(τ(x))
σ(x) (x)σ
( )
1 2 3 4
σ =
4 1 2 3
( )
1 2 3 4
τ =
.
2 1 4 3
στ =
τσ =
( )
1 2 3 4
,
1 4 3 2
( )
1 2 3 4
.
3 2 1 4
σ ∈ S X k a 1 ,a 2 ,...,a k ∈ X
σ(a 1 )=a 2

σ(a 2 )=a 3
σ(a k )=a 1
σ(x) =x x ∈ X (a 1 ,a 2 ,...,a k )
σ
σ =
6
( )
1 2 3 4 5 6 7
= (162354)
6 3 5 1 4 2 7
τ =
( )
1 2 3 4 5 6
= (243)
1 4 2 3 5 6
3
( )
1 2 3 4 5 6
= (1243)(56).
2 4 1 3 6 5
4
σ = (1352) τ = (256).
σ
τ
1 ↦→ 3, 3 ↦→ 5, 5 ↦→ 2, 2 ↦→ 1,
2 ↦→ 5, 5 ↦→ 6, 6 ↦→ 2,
στ τ σ
1 ↦→ 3, 3 ↦→ 5, 5 ↦→ 6, 6 ↦→ 2 ↦→ 1,
στ = (1356) μ = (1634) σμ = (1652)(34)
S X σ =(a 1 ,a 2 ,...,a k ) τ =(b 1 ,b 2 ,...,b l ) a i ≠ b j
i j
(135) (27) (135) (347)
(135)(27) = (135)(27)
(135)(347) = (13475).
σ τ S X στ = τσ

σ =(a 1 ,a 2 ,...,a k ) τ =(b 1 ,b 2 ,...,b l ) στ(x) =
τσ(x) x ∈ X x {a 1 ,a 2 ,...,a k } {b 1 ,b 2 ,...,b l } σ
τ x σ(x) =x τ(x) =x
στ(x) =σ(τ(x)) = σ(x) =x = τ(x) =τ(σ(x)) = τσ(x).
x ∈{a 1 ,a 2 ,...,a k }
σ(a i )=a (i k)+1
a 1 ↦→ a 2
a 2 ↦→ a 3
a k−1 ↦→ a k
a k ↦→ a 1 .
τ(a i )=a i σ τ
στ(a i )=σ(τ(a i ))
= σ(a i )
= a (i k)+1
= τ(a (i k)+1 )
= τ(σ(a i ))
= τσ(a i ).
x ∈{b 1 ,b 2 ,...,b l } σ τ
S n
X = {1, 2,...,n} σ ∈ S n X 1
{σ(1),σ 2 (1),...} X 1 X i
X X 1 X 2 {σ(i),σ 2 (i),...} X 2
X 3 ,X 4 ,... X
r σ i
{ σ(x) x ∈ Xi
σ i (x) =
x x /∈ X i ,
σ = σ 1 σ 2 ···σ r X 1 ,X 2 ,...,X r σ 1 ,σ 2 ,...,σ r
( )
1 2 3 4 5 6
σ =
6 4 3 1 5 2
( )
1 2 3 4 5 6
τ =
.
3 2 1 5 6 4
σ = (1624)

τ = (13)(456)
στ = (136)(245)
τσ = (143)(256).
(1)
(a 1 ,a 2 ,...,a n )=(a 1 a n )(a 1 a n−1 ) ···(a 1 a 3 )(a 1 a 2 ),
(16)(253) = (16)(23)(25) = (16)(45)(23)(45)(25).
(12)(12) (13)(24)(13)(24)
(16)
(23)(16)(23)
(35)(16)(13)(16)(13)(35)(56),
(16)
r
r
= τ 1 τ 2 ···τ r ,
r
r>1 r =2 r>2
τ r−1 τ r
(ab)(ab) =
(bc)(ab) =(ac)(bc)
(cd)(ab) =(ab)(cd)
(ac)(ab) =(ab)(bc),
a b c d
τ r−1 τ r
= τ 1 τ 2 ···τ r−3 τ r−2 .

− 2 r
τ r−1 τ r
r
a
τ r−2 τ r−1 r − 2
r a τ r−2
r − 2
τ r−3 τ r−2
a
a
r − 2
σ
σ
σ
σ
σ = σ 1 σ 2 ···σ m = τ 1 τ 2 ···τ n ,
m n σ σ m ···σ 1
= σσ m ···σ 1 = τ 1 ···τ n σ m ···σ 1 ,
n σ
S n A n
A n n
A n S n
A n A n σ
σ = σ 1 σ 2 ···σ r ,
σ i r
A n
σ −1 = σ r σ r−1 ···σ 1
S n n ≥ 2
A n n!/2

A n S n B n
σ S n n ≥ 2 σ
λ σ : A n → B n
λ σ (τ) =στ.
λ σ (τ) =λ σ (μ) στ = σμ
τ = σ −1 στ = σ −1 σμ = μ.
λ σ λ σ
A 4 S 4
A 4
(1) (12)(34) (13)(24) (14)(23)
(123) (132) (124) (142)
(134) (143) (234) (243).
A 4
n n n =3, 4,...
n D n
n 1, 2,...,n
n k
k +1 k −1 2n
n

n
1
2
n − 1 3
n
D n S n 2n
4
n
n
D n n ≥ 3 r
s
r n =1

s 2 =1
srs = r −1 .
n
n
, 360◦
n
, 2 · 360◦
n
360◦
,...,(n − 1) ·
n .
360 ◦ /n r r
r k = k · 360◦
n .
n s 1 ,s 2 ,...,s n s k k
n
s 1 = s n/2+1 ,s 2 = s n/2+2 ,...,s n/2 = s n
s 1 ,s 2 ,...,s n
s k s = s 1 s 2 =1 r n =1
t n k
k +1 k − 1 k +1 t = r k
k − 1 t = sr k r s D n
D n r s
D n = {1,r,r 2 ,...,r n−1 ,s,sr,sr 2 ,...,sr n−1 }.
srs = r −1
D 4
1 2 3 4
r = (1234)
r 2 = (13)(24)
r 3 = (1432)
r 4 =(1)
s 1 = (24)
s 2 = (13).
D 4 8
rs 1 = (12)(34)
r 3 s 1 = (14)(23).

D 4
n
6
6 · 4=24
24
S 4
24
S 4
1 2 3 4
S 4
180 ◦
S 4 S 4
S 4

( 1 2 3 4
) 5
2 4 1 5 3
( 1 2 3 4
) 5
3 5 1 4 2
( 1 2 3 4
) 5
4 2 5 1 3
( 1 2 3 4
) 5
1 4 3 2 5
(1345)(234)
(12)(1253)
(143)(23)(24)
(1423)(34)(56)(1324)
(1254)(13)(25)
(1254)(13)(25) 2
(1254) −1 (123)(45)(1254)
(1254) 2 (123)(45)
(123)(45)(1254) −2
(1254) 100
|(1254)|
|(1254) 2 |
(12) −1
(12537) −1
[(12)(34)(12)(47)] −1
[(1235)(467)] −1
(14356)
(156)(234)
(1426)(142)
(17254)(1423)(154632)
(142637)

(a 1 ,a 2 ,...,a n ) −1
S 4
{σ ∈ S 4 : σ(1) = 3}
{σ ∈ S 4 : σ(2) = 2}
{σ ∈ S 4 : σ(1) = 3 σ(2) = 2}
S 4
A 4
S 7 A 7
A 10 15
A 8 26
S n n =3,...,10
A 5 A 6
σ ∈ S n n i j σ i = σ j
i ≡ j ( n)
σ = σ 1 ···σ m ∈ S n σ
σ 1 ,...,σ m
D 5 r s
r s
(12)(34)
A 4
S n n ≥ 3
A n n ≥ 4
D n n ≥ 3
σ ∈ S n σ n − 1
σ ∈ S n σ σ
n − 2
σ
σ
σ σ 2
3
A n n ≥ 3 3
S n
(12), (13),...,(1n)
(12), (23),...,(n − 1,n)
(12), (12 ...n)

G λ g : G → G λ g (a) =ga λ g
G
n! n
G
Z(G) ={g ∈ G : gx = xg x ∈ G}.
D 8 D 10 D n
τ =(a 1 ,a 2 ,...,a k ) k
σ
k
στσ −1 =(σ(a 1 ),σ(a 2 ),...,σ(a k ))
μ k σ στσ −1 = μ
α β S n α ∼ β σ ∈ S n σασ −1 = β
∼ S n
σ ∈ S X σ n (x) =y x ∼ y
∼ X
σ ∈ A n τ ∈ S n τ −1 στ ∈ A n
x ∈ X σ ∈ S X
O x,σ = {y : x ∼ y}.
S 5
α = (1254)
β = (123)(45)
γ = (13)(25).
O x,σ ∩O y,σ ≠ ∅ O x,σ = O y,σ σ
∼
H S X x, y ∈ X σ ∈ H
σ(x) =y 〈σ〉 O x,σ = X x ∈ X
α ∈ S n n ≥ 3 αβ = βα β ∈ S n α
S n
α α −1 α
α −1 β −1 αβ α, β ∈ S n
r s D n
srs = r −1
r k s = sr −k D n
r k ∈ D n n/ (k, n)

6
G H G H
g ∈ G
gH = {gh : h ∈ H}.
Hg = {hg : h ∈ H}.
H Z 6 0 3
0+H =3+H = {0, 3}
1+H =4+H = {1, 4}
2+H =5+H = {2, 5}.
Z Z n
H S 3 {(1), (123), (132)}
H
(1)H = (123)H = (132)H = {(1), (123), (132)}
(12)H = (13)H = (23)H = {(12), (13), (23)}.
H
H(1) = H(123) = H(132) = {(1), (123), (132)}
H(12) = H(13) = H(23) = {(12), (13), (23)}.

K
S 3 {(1), (12)} K
K
(1)K = (12)K = {(1), (12)}
(13)K = (123)K = {(13), (123)}
(23)K = (132)K = {(23), (132)};
K(1) = K(12) = {(1), (12)}
K(13) = K(132) = {(13), (132)}
K(23) = K(123) = {(23), (123)}.
H G g 1 ,g 2 ∈ G
g 1 H = g 2 H
Hg −1
1 = Hg −1
2
g 1 H ⊂ g 2 H
g 2 ∈ g 1 H
g −1
1 g 2 ∈ H
H G
H G H G
G G H G
g 1 H g 2 H H G g 1 H ∩g 2 H =
∅ g 1 H = g 2 H g 1 H ∩ g 2 H ≠ ∅ a ∈ g 1 H ∩ g 2 H
a = g 1 h 1 = g 2 h 2 h 1 h 2 H g 1 = g 2 h 2 h −1
1
g 1 ∈ g 2 H g 1 H = g 2 H
G
H
G H G H G
H G [G : H]
G = Z 6 H = {0, 3} [G : H] =3
G = S 3 H = {(1), (123), (132)} K = {(1), (12)}
[G : H] =2 [G : K] =3
H G H G
H G

L H R H H G
φ : L H →R H gH ∈L H
φ(gH) =Hg −1 φ g 1 H = g 2 H
Hg1 −1 = Hg2 −1 φ
Hg −1
1 = φ(g 1 H)=φ(g 2 H)=Hg −1
2 .
g 1 H = g 2 H φ φ(g −1 H)=Hg
H G g ∈ G φ : H → gH
φ(h) =gh φ H
gH
φ φ(h 1 )=φ(h 2 )
h 1 ,h 2 ∈ H h 1 = h 2 φ(h 1 )=gh 1 φ(h 2 )=gh 2
gh 1 = gh 2 h 1 = h 2 φ
gH gh h ∈ H φ(h) =gh
G H G
|G|/|H| =[G : H] H G
H G
G [G : H]
|H| |G| =[G : H]|H|
G g ∈ G g
G
|G| = p p G g ∈ G
g ≠ e
g G g ≠ e g
|〈g〉| > 1 p g G
p Z p
H K G G ⊃ H ⊃ K
[G : K] =[G : H][H : K].
[G : K] = |G|
|K| = |G|
|H| · |H| =[G : H][H : K].
|K|
A 4
12 6
12 1 2 3
4 6
A 4 6
H A 4 3
H 3 H 3
6

A 4 6
[A 4 : H] =2 H A 4
H gH = Hg gHg −1 = H
g ∈ A 4 3 A 4 3 H
(123) H (123) −1 = (132)
H ghg −1 ∈ H g ∈ A 4 h ∈ H
(124)(123)(124) −1 = (124)(123)(142) = (243)
(243)(123)(243) −1 = (243)(123)(234) = (142)
H
(1), (123), (132), (243), (243) −1 = (234), (142), (142) −1 = (124).
A 4 6
τ μ S n
σ ∈ S n μ = στσ −1
τ =(a 1 ,a 2 ,...,a k )
μ =(b 1 ,b 2 ,...,b k ).
σ
σ(a 1 )=b 1
σ(a 2 )=b 2
σ(a k )=b k .
μ = στσ −1
τ =(a 1 ,a 2 ,...,a k ) k σ ∈ S n σ(a i )=b
σ(a (i k)+1 )=b ′ μ(b) =b ′
μ =(σ(a 1 ),σ(a 2 ),...,σ(a k )).
σ μ τ
φ φ : N → N φ(n) =1 n =1 n>1
φ(n) m 1 ≤ m

a n n>0 (a, n) =
1 a φ(n) ≡ 1( n)
U(n) φ(n) a φ(n) =1
a ∈ U(n) a φ(n) − 1 n a φ(n) ≡ 1( n)
n = p
φ(p) =p − 1
p
p ∤ a p a
a p−1 ≡ 1
( p).
b b p ≡ b ( p)
G g 5 h
7 |G| ≥35
G 60
G
〈8〉 Z 24
〈3〉 U(8)
3Z Z
A 4 S 4
A n S n
D 4 S 4
T C ∗
H = {(1), (123), (132)} S 4

SL 2 (R) GL 2 (R) SL 2 (R) GL 2 (R)
n =15 a =4
p =4n +3
x 2 ≡−1( p)
H G g 1 ,g 2 ∈ G
g 1 H = g 2 H
Hg1 −1 = Hg2
−1
g 1 H ⊂ g 2 H
g 2 ∈ g 1 H
g −1
1 g 2 ∈ H
ghg −1 ∈ H g ∈ G h ∈ H
gH = Hg g ∈ G
φ : L H →R H φ(gH) =Hg
g n = e g n
α, β ∈ S n
γ β = γαγ −1 β = γαγ −1 γ ∈ S n
α β
|G| =2n 2
G
[G : H] =2a b H ab ∈ H
[G : H] =2 gH = Hg
H K G gH ∩ gK H ∩ K
G
H K G ∼ G a ∼ b
h ∈ H k ∈ K hak = b
H = {(1), (123), (132)} A 4
G n φ(n)
G
n = p e 1
1 pe 2
2 ···pe k
k
p 1 ,p 2 ,...,p k
φ(n) =n
(1 − 1 )(1 − 1 )
···
(1 − 1 )
.
p 1 p 2 p k
n = ∑ d|n
φ(d)
n

7
... ...
A
B C B C A
B
C
A B C
C C
f
f −1 f
f −1 f

=00, =01,..., =
25
f(p) =p +3 26;
A ↦→ D, B ↦→ E,...,Z ↦→ C
f −1 (p) =p − 3 26 = p +23 26.
3, 14, 9, 7, 4, 20, 3.
0, 11, 6, 4, 1, 17, 0,
3
26
26
b f(p) =p + b 26
=04
=18
18 = 4 + b 26 b =14
f(p) =p +14 26.
f −1 (p) =p +12 26.
26
f(p) =ap + b 26.
f −1
c = ap + b 26

p a
(a, 26) = 1
f −1 (p) =a −1 p − a −1 b 26.
f(p) =ap + b 26
a ∈ Z 26
(a, 26) = 1 a =5 (5, 26) = 1
a −1 =21 f(p) =5p +3 26
3, 6, 7, 23, 8, 10, 3
f −1 (p) =21p − 21 · 3 26 = 21p +15 26.
p 1 p 2
=
(
p1
p 2
)
.
A 2 × 2 Z 26
f() =A + ,
Z 26
f −1 () =A −1 − A −1 .
7, 4, 11, 15
( ) 3 5
A = ,
1 2
( )
A −1 2 21
= .
25 3
=(2, 2)

f
f −1
p q n = pq
φ(n) =m =(p − 1)(q − 1) φ φ
E m E
(E,m)=1 D DE ≡ 1
( m) n E
E n
=00, =02,..., =25
n x y = x E n
y x x = y D n
D
25 p =23
q =29
n = pq = 667
φ(n) =m =(p − 1)(q − 1) = 616.
E = 487 (616, 487) = 1
25 487 667 = 169.
191E = 1+151m
(n, D) = (667, 191)
169 191 667 = 25.
DE ≡ 1( m)
k
DE = km +1=kφ(n)+1.

(x, n) =1
y D =(x E ) D = x DE = x km+1 =(x φ(n) ) k x =(1) k x = x n.
x y D n
(x, n) ≠1 n = pq x

2×2 A Z 26 ((A), 26) =
1
( ) 3 4
A = ,
2 3
f() = A +
=(2, 5)
x x
2 x = 142528 14 25 28
n = 3551,E = 629,x=31
n = 2257,E =47,x=23
n = 120979,E = 13251,x= 142371
n = 45629,E = 781,x= 231561
D
y
n = 3551,D = 1997,y = 2791
n = 5893,D =81,y =34
n = 120979,D = 27331,y = 112135
n = 79403,D = 671,y = 129381
D
(n, E)
(n, E) = (451, 231)
(n, E) = (3053, 1921)
(n, E) = (37986733, 12371)
(n, E) = (16394854313, 34578451)
n
n
n E X
X E ≡ X
( n).
10 15 (n, E) D

n n n
d =2, 3,..., √ n n d n
n
n = ab n
n = x 2 − y 2 =(x − y)(x + y).
x y n = x 2 − y 2
n
p
(a, p) =1 a p−1 ≡ 1( p)
15
17
2 15−1 ≡ 2 14 ≡ 4 ( 15).
2 17−1 ≡ 2 16 ≡ 1 ( 17).
n
2 n−1 ≡ 1 ( n).

342
811
561
771
631
n b (b, n) =1
b n−1 ≡ 1( n) n b 341
2 3
2000
2000
2000
561 = 3 · 11 · 17
2163 25 × 10 9

8
m
n
n
m
m n

n
n n
n (x 1 ,x 2 ,...,x n ) 3n
(x 1 ,x 2 ,...,x n ) ↦→ (x 1 ,x 2 ,...,x n ,x 1 ,x 2 ,...,x n ,x 1 ,x 2 ,...,x n ).
i i
(0110)
(0110 0110 0110)
(0110 1110 0110)
(0110)
n 2n
n m
m 3n
8 2 8 = 256 8
2 7 = 128
=65 10 = 01000001 2 ,
=66 10 = 01000010 2 ,
=67 10 = 01000011 2 .
128
00000000 2 =0 10 ,
01111111 2 = 127 10 .
0 1
1
= 01000001 2 ,
= 01000010 2 ,
n

= 11000011 2 .
(0100 0101)
1
m
8 16 32
0 1
1
1
1
0 1
(1001 1000)
000 001 010 011 100 101 110 111
000 0 1 1 2 1 2 2 3
111 3 2 2 1 2 1 1 0
0 1 0
(000) 1 (111)
(101)
1 0
(111)
(000) (111)
3

p q
pq
n
p
q
q
p
0 1 p
q =1− p
1 1 p
0 q
n p n p =0.999
(0.999) 10,000 ≈ 0.00005.
n (x 1 ,...,x n )
p
k
( n
q
k)
k p n−k .
k
k q
n − k p n
q k p n−k k
( n n!
=
k)
k!(n − k)! ,
n k
q k p n−k
( n
q
k)
k p n−k .

p =0.995 500
p n =(0.995) 500 ≈ 0.082.
( n
1)
qp n−1 = 500(0.005)(0.995) 499 ≈ 0.204.
( n
q
2)
2 p n−2 500 · 499
= (0.005) 2 (0.995) 498 ≈ 0.257.
2
1 − 0.082 − 0.204 − 0.257 = 0.457.
(n, m)
m n
(n, m)
E : Z m 2 → Z n 2
D : Z n 2 → Z m 2 .
E E
D
(8, 7)
E(x 7 ,x 6 ,...,x 1 )=(x 8 ,x 7 ,...,x 1 ),
x 8 = x 7 + x 6 + ···+ x 1 Z 2
=(x 1 ,...,x n ) =(y 1 ,...,y n ) n
d(, )
d
d(, )
w() 1 w() =d(, )
=(00···0)
= (10101) = (11010) = (00011)
C
d(, ) =4, d(, ) =3, d(, ) =3.
w() =3, w() =3, w() =2.

n
w() =d(, )
d(, ) ≥ 0
d(, ) =0 =
d(, ) =d(, )
d(, ) ≤ d(, )+d(, )
= (1101) = (1100)
(1101) (1100)
(1100)
d(, ) =1 = (1100)
= (1010) d(, ) =2
2
0000 0011 0101 0110 1001 1010 1100 1111
0000 0 2 2 2 2 2 2 4
0011 2 0 2 2 2 2 4 2
0101 2 2 0 2 2 4 2 2
0110 2 2 2 0 4 2 2 2
1001 2 2 2 4 0 2 2 2
1010 2 2 4 2 2 0 2 2
1100 2 4 2 2 2 2 0 2
1111 4 2 2 2 2 2 2 0
d(, ) =1
d(, ) =2
d =2
d(, ) =2
d(, ) =d(, ) =1.
d ≥ 3

d(, ) =1 d(, ) ≥ 2 ≠
3
C d =2n +1 C n
2n C
n
d(, ) ≤ n
2n +1≤ d(, ) ≤ d(, )+d(, ) ≤ n + d(, ).
d(, ) ≥ n +1
2n 1 ≤ d(, ) ≤ 2n 2n+1
1 2n
1 = (00000) 2 = (00111) 3 = (11100)
4 = (11011)
00000 00111 11100 11011
00000 0 3 3 4
00111 3 0 4 3
11100 3 4 0 3
11011 4 3 3 0
(32, 6)
Z n 2
n

n
(11000101) + (11000101) = (00000000).
(0000000) (0001111) (0010101) (0011010)
(0100110) (0101001) (0110011) (0111100)
(1000011) (1001100) (1010110) (1011001)
(1100101) (1101010) (1110000) (1111111).
Z 7 2
d =3
3
n w( + ) =d(, )
n
1
1+1=0
0+0=0
1+0=1
0+1=1.
d C d
C
d = {w() : ≠ }.
d = {d(, ) : ≠ }
= {d(, ) : + ≠ }
= {w( + ) : + ≠ }
= {w() : ≠ }.
3
n
· = x 1 y 1 + ···+ x n y n ,

=(x 1 ,x 2 ,...,x n ) =(y 1 ,y 2 ,...,y n )
= (011001) = (110101) · =0
· =
= ( x 1 x 2 ···
⎛ ⎞
y 1
)
y 2
x n ⎜ ⎟
⎝ ⎠
y n
= x 1 y 1 + x 2 y 2 + ···+ x n y n .
3
3
1 n =(x 1 ,x 2 ,...,x n )
1 x 1 + x 2 + ···+ x n =0 4 =(x 1 ,x 2 ,x 3 ,x 4 )
1 x 1 + x 2 + x 3 + x 4 =0
⎛ ⎞
1
· = = ( )
1
x 1 x 2 x 3 x 4 ⎜ ⎟
⎝1⎠ =0.
1
M m×n (Z 2 ) m × n Z 2
Z 2
H ∈ M m×n (Z 2 ) n
H = H (H)
⎛
0 1 0 1
⎞
0
H = ⎝1 1 1 1 0⎠ .
0 0 1 1 1
5 =(x 1 ,x 2 ,x 3 ,x 4 ,x 5 ) H H =
x 2 + x 4 =0
x 1 + x 2 + x 3 + x 4 =0
x 3 + x 4 + x 5 =0.
5
(00000) (11110) (10101) (01011).
H M m×n (Z 2 ) H
n

Z n 2
, ∈ (H) H M m×n (Z 2 ) H =
H =
H( + ) =H + H = + = .
+ H
H ∈
M m×n (Z 2 )
C
⎛
0 0 0 1 1
⎞
1
H = ⎝0 1 1 0 1 1⎠ .
1 0 1 0 0 1
6 = (010011)
⎛ ⎞
0
H = ⎝1⎠ ,
1
H H
H m × n Z 2 n>m m
m × m I m
H =(A | I m ) A m × (n − m)
⎛
⎞
a 11 a 12 ··· a 1,n−m
a 21 a 22 ··· a 2,n−m
⎜
⎝
⎟
⎠
a m1 a m2 ··· a m,n−m
I m m × m
⎛
1 0 ···
⎞
0
0 1 ··· 0
⎜
⎝
⎟
⎠ .
0 0 ··· 1
n × (n − m)
( )
In−m
G = .
A
G = H =
G

(000), (001), (010),...,(111).
⎛
0 1
⎞
1
A = ⎝1 1 0⎠ ,
1 0 1
⎛ ⎞
1 0 0
0 1 0
0 0 1
G =
0 1 1
⎜ ⎟
⎝1 1 0⎠
1 0 1
⎛
0 1 1 1 0
⎞
0
H = ⎝1 1 0 0 1 0⎠ ,
1 0 1 0 0 1
H
6 1 1
=(x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 6 )
⎛
⎞
x 2 + x 3 + x 4
= H = ⎝x 1 + x 2 + x 5
⎠ ,
x 1 + x 3 + x 6
x 2 + x 3 + x 4 =0
x 1 + x 2 + x 5 =0
x 1 + x 3 + x 6 =0.
x 4 x 2 x 3 x 5 x 1 x 2 x 6
x 1 x 3 x 4 x 5 x 6
x 1 x 2 x 3 x 4 x 5 x 6
H
(000000) (001101) (010110) (011011)
(100011) (101110) (110101) (111000).
G

G
000 000000
001 001101
010 010110
011 011011
100 100011
101 101110
110 110101
111 111000
H ∈ M m×n (Z 2 ) (H)
∈ Z n 2 n − m m
H = m
n − m H (n, n − m)
n − m m
G n × k C =
{
: G = ∈ Z
k
2
}
(n, k) C
G 1 = 1 G 2 = 2 1 + 2 C
G( 1 + 2 )=G 1 + G 2 = 1 + 2 .
G = G = G = G
G − G = G( − ) =.
k G( − ) x 1 − y 1 ,...,x k − y k
I k G G( − ) =
=
( )
H =(A | I m ) m×n G =
In−m
A
n × (n − m) HG =
C = HG ij C
n∑
c ij = h ik g kj
=
=
k=1
n−m
∑
h ik g kj +
k=1
n−m
∑
k=1
a ik δ kj +
n∑
k=n−m+1
n∑
k=n−m+1
h ik g kj
δ i−(m−n),k a kj

= a ij + a ij
=0,
δ ij =
{
1, i = j
0, i ≠ j
( ) H =(A | I m ) m × n
G =
In−m
A
n × (n − m) H C
G C H = C
H
∈ C G = ∈ Z m 2
H = HG =
=(y 1 ,...,y n ) H
Z n−m
2 G = H =
a 11 y 1 + a 12 y 2 + ···+ a 1,n−m y n−m + y n−m+1 =0
a 21 y 1 + a 22 y 2 + ···+ a 2,n−m y n−m + y n−m+1 =0
a m1 y 1 + a m2 y 2 + ···+ a m,n−m y n−m + y n−m+1 =0.
y n−m+1 ,...,y n y 1 ,...,y n−m
y n−m+1 = a 11 y 1 + a 12 y 2 + ···+ a 1,n−m y n−m
y n−m+1 = a 21 y 1 + a 22 y 2 + ···+ a 2,n−m y n−m
y n−m+1 = a m1 y 1 + a m2 y 2 + ···+ a m,n−m y n−m .
x i = y i i =1,...,n− m
H
1 = (100 ···00)
2 = (010 ···00)
n = (000 ···01)
n Z n 2 1 m × n H H i i
H
⎛ ⎞
0
⎛
⎞
1 1 1 0 0
⎜1
⎛ ⎞
⎟ 1
⎝
1 0 0 1 0
1 1 0 0 1
⎠
0
= ⎝0⎠ .
⎜ ⎟
⎝0⎠
1
0

i n 1 i 0
H ∈ M m×n (Z 2 ) H i i H
H m × n H
H
(H)
2
2
i i =1,...,n H i i
H i H i
H
H i ≠
⎛
1 1 1 0
⎞
0
H 1 = ⎝1 0 0 1 0⎠
1 1 0 0 1
⎛
1 1 1 0
⎞
0
H 2 = ⎝1 0 0 0 0⎠ ,
1 1 0 0 1
H 1 H 2
H H 2
3
⎛ ⎞
1 1 1 0
H = ⎝1 0 0 1⎠
1 1 0 0
H
(H) 4
2 (1100) (1010) (1001) (0110) (0101) (0011)
(H)
H H
H
H H
H H
n i + j 1 i j
w( i + j )=2 i ≠ j
= H( i + j )=H i + H j
i j H

H
2 3 =8
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
0 1 0 0
⎝ ⎠ , ⎝ ⎠ , ⎝ ⎠ , ⎝ ⎠ .
0
0
0
0
3
H m×n n−m
m 2 m
, 1 ,..., m 2 m − (1 + m)
1
0
0
1
n
n
⎛
1 1 1 0
⎞
0
H = ⎝0 1 0 1 0⎠
1 0 0 0 1
5 = (11011) = (01011)
⎛ ⎞
0
⎛ ⎞
1
H = ⎝0⎠ H = ⎝0⎠ .
0
1
H H
0 1
H m × n ∈ Z n 2 H
m × n H
n = +
H
H = H( + ) =H + H = + H = H.

H i
H
H ∈ M m×n (Z 2 ) H
n
H i i
⎛
1 0 1 1 0
⎞
0
H = ⎝0 1 1 0 1 0⎠
1 1 1 0 0 1
6 = (111110) = (111111) = (010111)
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1
1
1
H = ⎝1⎠ ,H = ⎝1⎠ ,H = ⎝0⎠ .
1
0
0
(110110) (010011)
H
C
Z n 2 C Zn 2
C (n, m) C
Z n 2 + C ∈ Zn 2
2 n−m C Z n 2
C (5, 3)
⎛
0 1 1 0
⎞
0
H = ⎝1 0 0 1 0⎠ .
1 1 0 0 1
(00000) (01101) (10011) (11110).
2 5−2 =2 3 C Z 5 2 22 =4

C (00000)(01101)(10011)(11110)
(10000) + C (10000)(11101)(00011)(01110)
(01000) + C (01000)(00101)(11011)(10110)
(00100) + C (00100)(01001)(10111)(11010)
(00010) + C (00010)(01111)(10001)(11100)
(00001) + C (00001)(01100)(10010)(11111)
(10100) + C (00111)(01010)(10100)(11001)
(00110) + C (00110)(01011)(10101)(11000)
C
n
= + = +
+ C
n
+
= (01111) (00010) + C
(01101) = (01111) + (00010)
(000) (00000)
(001) (00001)
(010) (00010)
(011) (10000)
(100) (00100)
(101) (01000)
(110) (00110)
(111) (10100)
C (n, k) H
Z n 2 C H = H
n
n C − ∈ C
H( − ) =0 H = H

C
= (01111)
⎛ ⎞
0
H = ⎝1⎠ .
1
(00010)
(n, k)
2 n−k C (32, 24)
2 24 2 32−24 =2 8 = 256
0 1 2 3 4 5 6 7 8
000 001 010 011 101 110 111 000 001
4 Z 4 2
(0110) (1001) (1010) (1100)
n
(011010), (011100)
(11110101), (01010100)
(00110), (01111)
(1001), (0111)
n
(011010)
(11110101)
(01111)
(1011)
C 7
C
(011010) (011100) (110111) (110000)
(011100) (011011) (111011) (100011)
(000000) (010101) (110100) (110011)
(000000) (011100) (110101) (110001)

(0110110) (0111100) (1110000) (1111111)
(1001001) (1000011) (0001111) (0000000)
(n, k)
⎛
0 1 0 0
⎞
0
⎝1 0 1 0 1⎠
1 0 0 1 0
( 1 0 0 1
) 1
0 1 0 1 1
⎛
⎞
1 0 1 0 0 0
1 1 0 1 0 0
⎜
⎟
⎝0 1 0 0 1 0⎠
1 1 0 0 0 1
⎛
⎞
0 0 0 1 1 1 1
0 1 1 0 0 1 1
⎜
⎟
⎝1 0 1 0 1 0 1⎠
0 1 1 0 0 1 1
(5, 2)
C
⎛
0 1 0 0
⎞
1
H = ⎝1 0 1 0 1⎠ .
0 0 1 1 1
01111 10101 01110 00011
1000
p
p =0.01
p =0.0001
⎛
⎞
1 1 0 0 0
0 0 1 0 0
⎜
⎟
⎝0 0 0 1 0⎠
1 0 0 0 1
⎛
⎞
0 1 1 0 0 0
1 1 0 1 0 0
⎜
⎟
⎝0 1 0 0 1 0⎠
1 1 0 0 0 1
( 1 1 1
) 0
1 0 0 1
⎛
⎞
0 0 0 1 0 0 0
0 1 1 0 1 0 0
⎜
⎟
⎝1 0 1 0 0 1 0⎠
0 1 1 0 0 0 1

⎛
⎞
0 1 1 1 1
H = ⎝0 0 0 1 1⎠ .
1 0 1 0 1
C Z 3 2 (000) (111)
C Z 3 2
C
⎛
0 1 0 0
⎞
0
⎝1 0 1 0 1⎠
1 0 0 1 0
( 1 0 0 1
) 1
0 1 0 1 1
⎛
⎞
0 0 1 0 0
1 1 0 1 0
⎜
⎟
⎝0 1 0 1 0⎠
1 1 0 0 1
⎛
⎞
1 0 0 1 1 1 1
1 1 1 0 0 1 1
⎜
⎟
⎝1 0 1 0 1 0 1⎠
1 1 1 0 0 1 0
n
w() =d(, )
d(, ) =d( + , + )
d(, ) =w( − )
X d : X × X → R
d(, ) ≥ 0 , ∈ X
d(, ) =0 =
d(, ) =d(, )
d(, ) ≤ d(, )+d(, )

Z n 2
C i C
C
C
i n 1 i 0
H ∈ M m×n (Z 2 ) H i i H
C (n, k) C
C ⊥ = { ∈ Z n 2 : · =0 ∈ C}.
C C
⎛
1 1 1 0
⎞
0
⎝0 0 1 0 1⎠ .
1 0 0 1 0
C ⊥ (n, n − k)
C C ⊥
H m × n Z 2 i i
m
⎛
0 0 0 1 1
⎞
1
H = ⎝0 1 1 0 0 1⎠
1 0 1 0 1 0
i i
(101011)

H
(101101) (001001)
(0010000101) (0000101100)
(m, n)
k
(16, 12)

9
Z 4
T i
(G, ·) (H, ◦)
φ : G → H
φ(a · b) =φ(a) ◦ φ(b)
a b G G H G ∼ = H φ
Z 4
∼ = 〈i〉 φ : Z4 →〈i〉 φ(n) =i n
φ φ
φ(0) = 1
φ(1) = i
φ(2) = −1
φ(3) = −i.
φ(m + n) =i m+n = i m i n = φ(m)φ(n),
φ
(R, +) (R + , ·)
φ(x + y) =e x+y = e x e y = φ(x)φ(y).
φ
Q ∗
2 n φ : Z → Q ∗ φ(n) =2 n
φ(m + n) =2 m+n =2 m 2 n = φ(m)φ(n).

φ {2 n : n ∈ Z} Q ∗
m ≠ n φ(m) ≠ φ(n)
m>n φ(m) =φ(n) 2 m =2 n 2 m−n =1
m − n>0
Z 8 Z 12
U(8) ∼ = U(12)
U(8) = {1, 3, 5, 7}
U(12) = {1, 5, 7, 11}.
φ : U(8) → U(12)
1 ↦→ 1
3 ↦→ 5
5 ↦→ 7
7 ↦→ 11.
φ
ψ ψ(1) = 1 ψ(3) = 11 ψ(5) = 5 ψ(7) = 7
Z 2 × Z 2
S 3 Z 6
Z 6 S 3
φ : Z 6 → S 3
a, b ∈ S 3 ab ≠ ba φ
m n Z 6
φ(m) =a φ(n) =b.
ab = φ(m)φ(n) =φ(m + n) =φ(n + m) =φ(n)φ(m) =ba,
a b
φ : G → H
φ −1 : H → G
|G| = |H|
G H
G H
G n H n
φ
h 1 h 2 H φ
g 1 ,g 2 ∈ G φ(g 1 )=h 1 φ(g 2 )=h 2
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
G a
G φ : Z → G φ : n ↦→ a n
φ(m + n) =a m+n = a m a n = φ(m)φ(n).
φ m n Z m ≠ n
m>n a m ≠ a n
a m = a n a m−n = e m − n>0
a G a n
n φ(n) =a n
G n G Z n
G n a φ : Z n → G
φ : k ↦→ a k 0 ≤ k < n φ
G p p G
Z p
G
G
Z 3 Z 3
+ 0 1 2
0 0 1 2
1 1 2 0
2 2 0 1
Z 3 G =
{(0), (012), (021)}
( ) 0 1 2
0 ↦→
=(0)
0 1 2

( ) 0 1 2
1 ↦→
= (012)
1 2 0
( ) 0 1 2
2 ↦→
= (021).
2 0 1
G G
G g ∈ G λ g : G → G λ g (a) =ga λ g
G λ g λ g (a) =λ g (b)
ga = λ g (a) =λ g (b) =gb.
a = b λ g a ∈ G b
λ g (b) =a b = g −1 a
G
G = {λ g : g ∈ G}.
G
G G
(λ g ◦ λ h )(a) =λ g (ha) =gha = λ gh (a).
λ e (a) =ea = a
(λ g −1 ◦ λ g )(a) =λ g −1(ga) =g −1 ga = a = λ e (a).
G G φ : g ↦→ λ g
φ(gh) =λ gh = λ g λ h = φ(g)φ(h).
φ(g)(a) =φ(h)(a)
ga = λ g a = λ h a = ha.
g = h φ φ(g) =λ g λ g ∈ G
g ↦→ λ g G

G H
G H G × H
G
G
(G, ·) (H, ◦) G H
(g, h) ∈ G × H g ∈ G
h ∈ H G × H
(g 1 ,h 1 )(g 2 ,h 2 )=(g 1 · g 2 ,h 1 ◦ h 2 );
G
H · ◦
(g 1 ,h 1 )(g 2 ,h 2 )=(g 1 g 2 ,h 1 h 2 )
G H G × H
(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
e G e H
G H (e G ,e H ) G × H
(g, h) ∈ G × H (g −1 ,h −1 )
G H
R
R R × R = R 2
(a, b) +(c, d) =(a + c, b + d) (0, 0)
(a, b) (−a, −b)
Z 2 × Z 2 = {(0, 0), (0, 1), (1, 0), (1, 1)}.
Z 2 × Z 2 Z 4
(a, b) Z 2 × Z 2 2 (a, b)+(a, b) =(0, 0) Z 4
G × H G H
n∏
G i = G 1 × G 2 ×···×G n
i=1
G 1 ,G 2 ,...,G n G = G 1 = G 2 =
···= G n G n G 1 × G 2 ×···×G n
Z n 2 n
n
(01011101) + (01001011) = (00010110).

(g, h) ∈ G × H g h r s
(g, h) G × H r s
m r s n = |(g, h)|
(g, h) m =(g m ,h m )=(e G ,e H )
(g n ,h n )=(g, h) n =(e G ,e H ).
n m n ≤ m r s
n n r s m
r s m ≤ n m n
(g 1 ,...,g n ) ∈ ∏ G i g i r i G i
(g 1 ,...,g n ) ∏ G i r 1 ,...,r n
(8, 56) ∈ Z 12 × Z 60 (8, 12) = 4 8 12/4 = 3
Z 12 56 Z 60 15 3 15 15
(8, 56) 15 Z 12 × Z 60
Z 2 × Z 3
(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2).
Z 2 × Z 2 Z 4 Z 2 × Z 3
∼ = Z6
Z 2 × Z 3 (1, 1) Z 2 × Z 3
Z m × Z n Z mn (m, n) =1
Z m × Z n
∼ = Zmn (m, n) =1
(m, n) =d>1 Z m ×Z n
mn/d m n (a, b) ∈ Z m × Z n
(a, b)+(a, b)+···+(a, b) =(0, 0).
} {{ }
mn/d
(a, b) Z m × Z n
(m, n) =mn
(m, n) =1
n 1 ,...,n k
(n i ,n j )=1 i ≠ j
k∏
Z ni
∼ = Zn1···n k
i=1
m = p e 1
1 ···pe k
k
,
p i
Z m
∼ = Zp e 1
1
×···×Z p
e k
k
.

p e i
i
p e j
j
i ≠ j
Z e p 1 ×···×Z e
1 p k
k
p 1 ,...,p k
G H K
G = HK = {hk : h ∈ H, k ∈ K}
H ∩ K = {e}
hk = kh k ∈ K h ∈ H
G H K
U(8)
H = {1, 3} K = {1, 5}.
D 6
H = {,r 3 } K = {,r 2 ,r 4 ,s,r 2 s, r 4 s}.
K ∼ = S 3 D 6
∼ = Z2 × S 3
S 3
H K H 3 H
{(1), (123), (132)} K 2
K hk = kh h ∈ H
k ∈ K
G H K G
H × K
G g ∈ G g = hk
h ∈ H k ∈ K φ : G → H × K φ(g) =(h, k)
φ
h k g g = hk = h ′ k ′
h −1 h ′ = k(k ′ ) −1 H K h = h ′
k = k ′ φ
φ g 1 = h 1 k 1 g 2 = h 2 k 2
φ(g 1 g 2 )=φ(h 1 k 1 h 2 k 2 )
= φ(h 1 h 2 k 1 k 2 )

=(h 1 h 2 ,k 1 k 2 )
=(h 1 ,k 1 )(h 2 ,k 2 )
= φ(g 1 )φ(g 2 ).
φ
Z 6 {0, 2, 4}×{0, 3}
G
H 1 ,H 2 ,...,H n G
G = H 1 H 2 ···H n = {h 1 h 2 ···h n : h i ∈ H i }
H i ∩〈∪ j≠i H j 〉 = {e}
h i h j = h j h i h i ∈ H i h j ∈ H j
G H i i =1, 2,...,n
G ∏ i H i
Z ∼ = nZ n ≠0
C ∗ GL 2 (R)
( ) a b
.
−b a
U(8) ∼ = Z 4
U(8)
( ) ( ) ( ) ( )
1 0 1 0 −1 0 −1 0
, , ,
.
0 1 0 −1 0 1 0 −1
U(5) U(10) U(12)
n Z n
n Z n
Q Z
G = R \{−1} G
a ∗ b = a + b + ab.
G (G, ∗)
⎛
1 0
⎞
0
⎛
1 0
⎞
0
⎛
0 1
⎞
0
⎝0 1 0⎠
⎝0 0 1⎠
⎝1 0 0⎠
0 0 1 0 1 0 0 0 1

⎛
0 0
⎞
1
⎝1 0 0⎠
0 1 0
⎛
0 0
⎞
1
⎝0 1 0⎠
1 0 0
⎛
0 1
⎞
0
⎝0 0 1⎠
1 0 0
G 6
8
S 4 D 12
ω = (2π/n) n
A =
( ) ω 0
0 ω −1
B =
D n
( ) ±1 k
,
0 1
( ) 0 1
1 0
D n Z n
Z 4 × Z 2
(3, 4) Z 4 × Z 6
(6, 15, 4) Z 30 × Z 45 × Z 24
(5, 10, 15) Z 25 × Z 25 × Z 25
(8, 8, 8) Z 10 × Z 24 × Z 80
D 4
Q ∗ 2 m 3 n m, n ∈ Z
Z × Z
S 3 × Z 2 D 6 D 2n
3
3
6
G 20 G H K 4 5
hk = kh h ∈ H k ∈ K G
H K
G H K G × K ∼ =
H × K G ∼ = H
51
52
φ : G → H φ(x) =e H x = e G
e G e H G H
G ∼ = H G H
G p p Z p

S n A n+2
D n S n
φ : G 1 → G 2 ψ : G 2 → G 3 φ −1 ψ ◦ φ
U(5) ∼ = Z 4 U(p) p
S 3
G
φ(a + bi) =a − bi C C
a + ib ↦→ a − ib C ∗
A ↦→ B −1 AB SL 2 (R) B GL 2 (R)
G (G) (G)
S G G
(Z 6 )
(Z)
G H (G) ∼ = (H)
G g ∈ G i g : G → G i g (x) = gxg −1
i g G
(G)
(G) (G)
Q 8 (G) =(G)
G g ∈ G λ g : G → G ρ g : G → G λ g (x) =gx
ρ g (x) =xg −1 i g = ρ g ◦ λ g G
g ↦→ ρ g G
G H K
φ : G → H × K φ(g) =(h, k) g = hk h ∈ H k ∈ K
G H G n H
n
G ∼ = G H ∼ = H G × H ∼ = G × H
G × H H × G
n 1 ,...,n k
k∏
Z ni
∼ = Zn1···n k
i=1
(n i ,n j )=1 i ≠ j
A × B A B
∏
G H 1 ,H 2 ,...,H n G
i H i
H 1 H 2 G 1 G 2 H 1 × H 2
G 1 × G 2

m, n ∈ Z 〈m, n〉 = 〈d〉 d = (m, n)
m, n ∈ Z 〈m〉∩〈n〉 = 〈l〉 l = (m, n)
2p
2p p
G 2p p a ∈ G a
1 2 p 2p
G 2p G Z 2p
G
G 2p G
p G p
G 2p G
2
P G p y ∈ G 2 yP = Py
G 2p P = 〈z〉
p z y 2 yz = z k y
2 ≤ k

10
H G
gH = Hg g ∈ G
H G gH = Hg g ∈ G
G
G H G
gh = hg g ∈ G h ∈ H gH = Hg
H S 3 (1) (12)
(123)H = {(123), (13)} H(123) = {(123), (23)},
H S 3 N
(1) (123) (132) N
N = {(1), (123), (132)}
(12)N = N(12) = {(12), (13), (23)}.
G N G
N G
g ∈ G gNg −1 ⊂ N
g ∈ G gNg −1 = N

⇒ N G gN = Ng g ∈ G
g ∈ G n ∈ N n ′ N gn = n ′ g gng −1 = n ′ ∈ N
gNg −1 ⊂ N
⇒ g ∈ G gNg −1 ⊂ N N ⊂ gNg −1 n ∈ N
g −1 ng = g −1 n(g −1 ) −1 ∈ N g −1 ng = n ′ n ′ ∈ N n = gn ′ g −1
gNg −1
⇒ gNg −1 = N g ∈ G n ∈ N
n ′ ∈ N gng −1 = n ′ gn = n ′ g gN ⊂ Ng
Ng ⊂ gN
N G N G G/N
(aN)(bN) =abN G
N G/N
N G N G
G/N [G : N]
G/N (aN)(bN) =abN
aN = bN cN = dN
(aN)(cN) =acN = bdN =(bN)(dN).
a = bn 1 c = dn 2 n 1 n 2 N
acN = bn 1 dn 2 N
= bn 1 dN
= bn 1 Nd
= bNd
= bdN.
eN = N g −1 N
gN G/N N G
S 3 N = {(1), (123), (132)}
N S 3 N (12)N S 3 /N
N (12)N
N N (12)N
(12)N (12)N N
Z 2
S 3 /N
S 3 N = A 3
(12)N = {(12), (13), (23)}
G/N

3Z Z 3Z Z
0+3Z = {...,−3, 0, 3, 6,...}
1+3Z = {...,−2, 1, 4, 7,...}
2+3Z = {...,−1, 2, 5, 8,...}.
Z/3Z
+ 0+3Z 1+3Z 2+3Z
0+3Z 0+3Z 1+3Z 2+3Z
1+3Z 1+3Z 2+3Z 0+3Z
2+3Z 2+3Z 0+3Z 1+3Z
nZ Z Z/nZ
nZ
1+nZ
2+nZ
(n − 1) + nZ.
k + Z l + Z k + l + Z
D n r s
r n =
s 2 =
srs = r −1 .
r R n D n srs −1 =
srs = r −1 ∈ R n D n D n /R n
Z 2
Z p
p
A n
n ≥ 5
A n 3 n ≥ 3
3 A n
3 (ab) =(ba)
(ab)(ab) =

(ab)(cd) =(acb)(acd)
(ab)(ac) =(acb).
N A n n ≥ 3 N 3
N = A n
A n 3 (ijk)
i j {1, 2,...,n} k 3
3
(iaj) =(ija) 2
(iab) =(ijb)(ija) 2
(jab)=(ijb) 2 (ija)
(abc) =(ija) 2 (ijc)(ijb) 2 (ija).
N A n n ≥ 3 N
3 (ija) N
[(ij)(ak)](ija) 2 [(ij)(ak)] −1 =(ijk)
N N 3 (ijk) 1 ≤ k ≤ n
3 A n N = A n
n ≥ 5 N A n 3
σ N
σ
σ 3
σ σ = τ(a 1 a 2 ···a r ) ∈ N r>3
σ σ = τ(a 1 a 2 a 3 )(a 4 a 5 a 6 )
σ = τ(a 1 a 2 a 3 ) τ
σ = τ(a 1 a 2 )(a 3 a 4 ) τ
σ 3 N σ
σ = τ(a 1 a 2 ···a r )
N N
N
(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 2 a 3 ) −1
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 )
=(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 )
=(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 )
=(a 1 a 3 a r ),

N 3 N = A n
N
σ = τ(a 1 a 2 a 3 )(a 4 a 5 a 6 ).
σ −1 (a 1 a 2 a 4 )σ(a 1 a 2 a 4 ) −1 ∈ N
(a 1 a 2 a 4 )σ(a 1 a 2 a 4 ) −1 ∈ N.
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
=(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 )
=(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 )
=(a 1 a 4 a 2 a 6 a 3 ).
N
N σ = τ(a 1 a 2 a 3 ) τ
σ ∈ N σ 2 ∈ N
σ 2 = τ(a 1 a 2 a 3 )τ(a 1 a 2 a 3 )
=(a 1 a 3 a 2 ).
N 3
σ = τ(a 1 a 2 )(a 3 a 4 ),
τ 2
σ −1 (a 1 a 2 a 3 )σ(a 1 a 2 a 3 ) −1
N (a 1 a 2 a 3 )σ(a 1 a 2 a 3 ) −1 N
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
=(a 1 a 3 )(a 2 a 4 ).
n ≥ 5 b ∈{1, 2,...,n} b ≠ a 1 ,a 2 ,a 3 ,a 4 μ =(a 1 a 3 b)
μ −1 (a 1 a 3 )(a 2 a 4 )μ(a 1 a 3 )(a 2 a 4 ) ∈ N
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 )
=(a 1 a 3 b).
N 3
A n n ≥ 5
N A n N 3
N = A n A n
n ≥ 5

A 5
196,833 × 196,833
G H G
H G/H
G = S 4 H = A 4
G = A 5 H = {(1), (123), (132)}
G = S 4 H = D 4
G = Q 8 H = {1, −1,I,−I}
G = Z H =5Z
D 4
D 4
Q 8
Q 8
T 2 × 2 R
( ) a b
,
0 c
a b c ∈ R ac ≠0 U
( ) 1 x
,
0 1
x ∈ R

U T
U
U T
T /U
T GL 2 (R)
G G/H
H G H G/H
G
G G/H
H G/H G
H 2 G H
G S n n ≥ 3
G H k H G
g G
C(g) ={x ∈ G : xg = gx}.
C(g) G g G C(g)
G
G
S 3
GL 2 (R)
Z(G) ={x ∈ G : xg = gx g ∈ G}.
G G
G/Z(G) G
G G ′ = 〈aba −1 b −1 〉 G ′
G aba −1 b −1 G ′
G
G ′ G
N G G/N N
G

11
(G, ·) (H, ◦) φ : G → H
φ(g 1 · g 2 )=φ(g 1 ) ◦ φ(g 2 )
g 1 ,g 2 ∈ G φ H φ
S n
Z 2 S n
Z 2
G g ∈ G φ : Z → G φ(n) =g n
φ
φ(m + n) =g m+n = g m g n = φ(m)φ(n).
Z G g
G = GL 2 (R)
( ) a b
A =
c d
G (A) =ad − bc ≠0
A B G (AB) =(A) (B)
φ : GL 2 (R) → R ∗ A ↦→ (A)

T z
|z| =1 φ R T
φ : θ ↦→ θ + i θ
φ(α + β) =(α + β)+i (α + β)
=( α β − α β)+i( α β + α β)
=( α + i α)( β + i β)
= φ(α)φ(β).
φ : G 1 → G 2
e G 1 φ(e) G 2
g ∈ G 1 φ(g −1 )=[φ(g)] −1
H 1 G 1 φ(H 1 ) G 2
H 2 G 2 φ −1 (H 2 )={g ∈ G 1 : φ(g) ∈ H 2 } G 1
H 2 G 2 φ −1 (H 2 ) G 1
e e ′ G 1 G 2
e ′ φ(e) =φ(e) =φ(ee) =φ(e)φ(e).
φ(e) =e ′
φ(g −1 )φ(g) =φ(g −1 g)=φ(e) =e ′ .
φ(H 1 ) G 2 φ(H 1 ) H 1
G 1 x y φ(H 1 ) a, b ∈ H 1
φ(a) =x φ(b) =y
xy −1 = φ(a)[φ(b)] −1 = φ(ab −1 ) ∈ φ(H 1 ),
φ(H 1 ) G 2
H 2 G 2 H 1 φ −1 (H 2 ) H 1
g ∈ G 1 φ(g) ∈ H 2 H 1 φ(e) =e ′ a b H 1
φ(ab −1 )=φ(a)[φ(b)] −1 H 2 H 2 G 2 ab −1 ∈ H 1
H 1 G 1 H 2 G 2 g −1 hg ∈ H 1
h ∈ H 1 g ∈ G 1
φ(g −1 hg) =[φ(g)] −1 φ(h)φ(g) ∈ H 2 ,
H 2 G 2 g −1 hg ∈ H 1
φ : G → H e H
φ −1 ({e}) G φ
φ G
H

φ : G → H φ
G
φ : GL 2 (R) → R ∗ A ↦→
(A) 1 R ∗ 2 × 2
φ = SL 2 (R)
φ : R → C ∗ φ(θ) =
θ + i θ {2πn : n ∈ Z} φ ∼ = Z
φ
Z 7 Z 12 φ Z 7
{0} Z 7 Z 7 Z 12
Z 12
7 Z 7 Z 12
G g ∈ G φ Z
G φ(n) =g n g
{0} φ Z G g
g n φ nZ
φ : G → H G φ
G
H G
φ : G → G/H
φ(g) =gH.
φ(g 1 g 2 )=g 1 g 2 H = g 1 Hg 2 H = φ(g 1 )φ(g 2 ).
H
ψ : G → H
K = ψ K G φ : G → G/K
η : G/K → ψ(G) ψ = ηφ
K G η : G/K → ψ(G) η(gK) =
ψ(g) η g 1 K = g 2 K k ∈ K
g 1 k = g 2
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)
ψ = ηφ η
η(g 1 Kg 2 K)=η(g 1 g 2 K)
= ψ(g 1 g 2 )
= ψ(g 1 )ψ(g 2 )
= η(g 1 K)η(g 2 K).
η ψ(G) η η(g 1 K)=η(g 2 K)
ψ(g 1 )=ψ(g 2 ) ψ(g1 −1 g 2)=e g1 −1 g 2 ψ
g1 −1 g 2K = K g 1 K = g 2 K
ψ = ηφ
G
ψ
H
φ
η
G/K
G g φ : Z → G
n ↦→ g n
φ(m + n) =g m+n = g m g n = φ(m)φ(n).
φ |g| = m g m = e φ = mZ Z/ φ = Z/mZ ∼ = G
g φ =0 φ
G Z
Z Z n
H G
G N G HN
G H ∩ N H
H/H ∩ N ∼ = HN/N.
HN = {hn : h ∈ H, n ∈ N} G
h 1 n 1 ,h 2 n 2 ∈ HN N (h 2 ) −1 n 1 h 2 ∈ N
(h 1 n 1 )(h 2 n 2 )=h 1 h 2 ((h 2 ) −1 n 1 h 2 )n 2
HN hn ∈ HN HN
(hn) −1 = n −1 h −1 = h −1 (hn −1 h −1 ).

H ∩ N H h ∈ H n ∈ H ∩ N
h −1 nh ∈ H H h −1 nh ∈ N N G
h −1 nh ∈ H ∩ N
φ H HN/N h ↦→ hN φ
hnN = hN h H φ
φ(hh ′ )=hh ′ N = hNh ′ N = φ(h)φ(h ′ ).
φ H/ φ
HN/N = φ(H) ∼ = H/ φ.
φ = {h ∈ H : h ∈ N} = H ∩ N,
HN/N = φ(H) ∼ = H/H ∩ N
N G
H ↦→ H/N H
N G/N G N
G/N
H G N N H H/N
aN bN H/N (aN)(b −1 N)=ab −1 N ∈ H/N
H/N G/N
S G/N N H = {g ∈ G :
gN ∈ S} h 1 ,h 2 ∈ H (h 1 N)(h 2 N)=h 1 h 2 N ∈ S h −1
1 N ∈ S
H G H N S = H/N
H ↦→ H/N
H 1 H 2 G N H 1 /N = H 2 /N
h 1 ∈ H 1 h 1 N ∈ H 1 /N h 1 N = h 2 N ⊂ H 2 h 2 H 2
N H 2 h 1 ∈ H 2 H 1 ⊂ H 2 H 2 ⊂ H 1
H 1 = H 2 H ↦→ H/N
H G N H
G/N → G/H gN ↦→ gH
H/N H/N G/N
H/N G/N
G → G/N → G/N
H/N
H H G
G N H
G N ⊂ H
G/H ∼ = G/N
H/N .
Z/mZ ∼ = (Z/mnZ)/(mZ/mnZ).
|Z/mnZ| = mn |Z/mZ| = m |mZ/mnZ| = n

(AB) = (A) (B) A, B ∈ GL 2 (R)
GL 2 (R) R ∗
φ : R ∗ → GL 2 (R)
φ : R → GL 2 (R)
φ : GL 2 (R) → R
φ : GL 2 (R) → R ∗
φ(a) =
φ(a) =
( ) 1 0
0 a
( ) 1 0
a 1
(( )) a b
φ
= a + d
c d
(( )) a b
φ
= ad − bc
c d
φ : M 2 (R) → R
(( )) a b
φ
= b,
c d
M 2 (R) 2 × 2 R
A m × n x ↦→ Ax
φ : R n → R m
φ : Z → Z φ(n) =7n φ
φ
Z 24 Z 18
Z Z 12
Z 24 H = 〈4〉 N = 〈6〉
HN H + N
H ∩ N
HN/N
H/(H ∩ N)

HN/N H/(H ∩ N)
G n ∈ N φ : G → G g ↦→ g n
φ : G → H G φ(G)
φ : G → H G φ(G)
G H k H G
Q/Z ∼ = Q
G N G H G/N
φ −1 (H) G |H|·|N| φ : G → G/N
G 1 G 2 H 1 H 2 G 1
G 2 φ : G 1 → G 2 φ
φ :(G 1 /H 1 ) → (G 2 /H 2 ) φ(H 1 ) ⊂ H 2
H K G H ∩ K = {e} G
G/H × G/K
φ : G 1 → G 2 H 1
G 1 φ(H 1 )=H 2 G 1 /H 1
∼ = G2 /H 2
φ : G → H φ
φ −1 (e) ={e}
φ : G → H ∼ G a ∼ b φ(a) =φ(b)
a, b ∈ G
(G) G G
G
(G) ≤ S G
G
i g : G → G,
i g (x) =gxg −1 ,
g ∈ G i g ∈ (G)
(G) (G)
(G)
G
G i g G
G → (G)

g ↦→ i g .
(G) Z(G)
G/Z(G) ∼ = (G).
(S 3 ) (S 3 ) D 4
φ : Z → Z (Z)
Z 8 (Z 8 ) ∼ = U(8)
k ∈ Z n φ k : Z n → Z n a ↦→ ka φ k
φ k k Z n
Z n φ k k Z n
ψ : U(n) → (Z n ) ψ : k ↦→ φ k

12
T : R n → R m
R n α ∈ R
T ( + ) =T ()+T ()
T (α) =αT ().
m × n R R n R m
=(x 1 ,...,x n ) =(y 1 ,...,y n ) R n
m × n
⎛
⎞
a 11 a 12 ··· a 1n
a 21 a 22 ··· a 2n
A = ⎜
⎝
⎟
⎠
a m1 a m2 ··· a mn
R m α
A( + ) =A + A αA = A(α),
⎛ ⎞
x 1
x 2
= ⎜ ⎟
⎝ ⎠ .
x n

A (a ij )
T : R n → R m A T
T
1 =(1, 0,...,0)
2 =(0, 1,...,0)
n =(0, 0,...,1) .
=(x 1 ,...,x n )
x 1 1 + x 2 2 + ···+ x n n .
T ( 1 )=(a 11 ,a 21 ,...,a m1 ) ,
T ( 2 )=(a 12 ,a 22 ,...,a m2 ) ,
T ( n )=(a 1n ,a 2n ,...,a mn ) ,
T () =T (x 1 1 + x 2 2 + ···+ x n n )
= x 1 T ( 1 )+x 2 T ( 2 )+···+ x n T ( n )
( n∑
)
n∑
= a 1k x k ,..., a mk x k
k=1
k=1
= A.
T : R 2 → R 2
T (x 1 ,x 2 )=(2x 1 +5x 2 , −4x 1 +3x 2 ),
T
T 1 =(2, −4) T 2 =(5, 3) T
A =
( ) 2 5
.
−4 3
n × n A
A −1 AA −1 = A −1 A = I
⎛
⎞
1 0 ··· 0
0 1 ··· 0
I = ⎜
⎝
⎟
⎠
0 0 ··· 1
n × n A
A

A
A
A −1 =
( ) 2 1
,
5 3
( ) 3 −1
.
−5 2
A −1 (A) =2· 3 − 5 · 1=1
A B n × n
(AB) =( A)( B)
A (A −1 )=1/ A
A =(a ij ) A =(a ji ) (A )= A
T n × n A T
| A| R 2 T
| A|
2 × 2
n × n
GL n (R)
(A) =1 (B) =1 (AB) =(A) (B) =1 (A −1 )=
1/ A =1 SL n (R)
2 × 2
( ) a b
A = ,
c d
A ad − bc GL 2 (R)
ad − bc ≠0 A
A −1 =
1
ad − bc
( d −b
−c a
)
.
A SL 2 (R)
( )
A −1 d −b
=
.
−c a

SL 2 (R)
A =
( ) 1 1
0 1
SL 2 (R) =(1, 0)
=(0, 1) A (1, 0) (1, 1)
A =(1, 0) A =(1, 1)
y
y
(0, 1)
(1, 1)
(1, 0)
x
(1, 0)
x
SL 2 (R)
O(n)
GL n (R) A A −1 =
A O(n)
n × n O(n)
GL n (R)
( )
3/5 −4/5
,
4/5 3/5
( √ )
1/2 − 3/2
√ ,
3/2 1/2
⎛
−1/ √ 2 0 1/ √ ⎞
2
⎝ 1/ √ 6 −2/ √ 6 1/ √ 6
1/ √ 3 1/ √ 3 1/ √ ⎠ .
3
O(n)
=(x 1 ,...,x n ) =(y 1 ,...,y n )
⎛ ⎞
y 1
〈, 〉 = y 2
=(x 1 ,x 2 ,...,x n ) ⎜ ⎟
⎝ ⎠ = x 1y 1 + ···+ x n y n .
y n
=(x 1 ,...,x n )
‖‖ = √ √
〈, 〉 = x 2 1 + ···+ x2 n.
‖ − ‖

R n α ∈ R
〈, 〉 = 〈, 〉
〈, + 〉 = 〈, 〉 + 〈, 〉
〈α, 〉 = 〈,α〉 = α〈, 〉
〈, 〉 ≥0 =0
〈, 〉 =0 R n =0
=(3, 4) √ 3 2 +4 2 =5
( )
3/5 −4/5
A =
4/5 3/5
A =(−7/5, 24/5)
(AA )=(I) =1 (A) =(A )
1 −1
⎛ ⎞
a 1j
a 2j
j = ⎜ ⎟
⎝ ⎠
a nj
A =(a ij ) AA = I 〈 r , s 〉 = δ rs
δ rs =
{ 1 r = s
0 r ≠ s
n×n
A A −1 = A
A
‖T − T ‖ = ‖ − ‖ ‖T ‖ = ‖‖ 〈T ,T〉 = 〈, 〉
A n × n
A
A −1 = A
〈A,A〉 = 〈, 〉
‖A − A‖ = ‖ − ‖
‖A‖ = ‖‖

(2) ⇒ (3)
(3) ⇒ (2)
〈A,A〉 =(A) A
= A A
=
= 〈, 〉.
〈, 〉 = 〈A,A〉
= A A
= 〈,A A〉,
〈, (A A − I)〉 =0 A A − I =0 A −1 = A
(3) ⇒ (4) A A
‖A − A‖ 2 = ‖A( − )‖ 2
= 〈A( − ),A( − )〉
= 〈 − , − 〉
= ‖ − ‖ 2 .
(4) ⇒ (5) A A =0
‖A‖ = ‖A − A‖ = ‖ − ‖ = ‖‖.
(5) ⇒ (3)
〈, 〉 = 1 [
‖ + ‖ 2 −‖‖ 2 −‖‖ 2] .
2
〈A,A〉 = 1 [
‖A + A‖ 2 −‖A‖ 2 −‖A‖ 2]
2
= 1 [
‖A( + )‖ 2 −‖A‖ 2 −‖A‖ 2]
2
= 1 [
‖ + ‖ 2 −‖‖ 2 −‖‖ 2]
2
= 〈, 〉.
y
y
( θ, − θ)
(a, b)
(a, −b)
x
θ
( θ, θ)
x
O(2) R 2

R 2
T ∈ O(2) 1 =(1, 0) 2 =(0, 1) T ( 1 )=(a, b)
a 2 + b 2 =1 T ( 2 )=(−b, a) T
( ) ( )
a −b θ − θ
A = =
,
b a θ θ
0 ≤ θ

R n T () = +
T O(n)
R 2 R 2
R n
y
y
x
x
T ()
f R 2
f O(2)
f R 2 f
f(0) = 0 ‖f()‖ = ‖‖
‖‖ 2 − 2〈f(),f()〉 + ‖‖ 2 = ‖f()‖ 2 − 2〈f(),f()〉 + ‖f()‖ 2
= 〈f() − f(),f() − f()〉
= ‖f() − f()‖ 2
= ‖ − ‖ 2
= 〈 − , − 〉
= ‖‖ 2 − 2〈, 〉 + ‖‖ 2 .
〈f(),f()〉 = 〈, 〉.
1 2 (1, 0) (0, 1)
=(x 1 ,x 2 )=x 1 1 + x 2 2 ,
f() =〈f(),f( 1 )〉f( 1 )+〈f(),f( 2 )〉f( 2 )=x 1 f( 1 )+x 2 f( 2 ).
f
f T f R 2
T f() =A A ∈ O(2) f() =A+
f() =A + 1
g() =B + 2 ,

f(g()) = f(B + 2 )=AB + A 2 + 1 .
R 2 E(2)
R 2 E(2)
R n R n
X ⊂ R 2 X
R n X R 1 Z 2
R 2 O(2)
R 2 Z n D n
G = {f 1 ,f 2 ,...,f n }
X ⊂ R 2 ∈ X
G X S = { 1 , 2 ,... n } i = f i ()
= 1 n
n∑
i .
X
G R 2
O(2)
O(2)
( )
θ − θ
R θ =
θ θ
i=1
( )( )
φ − φ 1 0
T φ =
φ φ 0 −1
( )
φ φ
=
.
φ − φ
(R θ )=1 (T φ )=−1 Tφ 2 = I
G
G −1
G G
G θ 0
R θ0 R θ0
G n θ 1 nθ 0 (n +1)θ 0
(n +1)θ 0 − θ 1 θ 0
θ 0
G T φ : G →
{−1, 1} A ↦→ (A)
|G/ φ| =2 G
n |G| =2n G
R θ ,...,R n−1
θ
,TR θ ,...,TR n−1
θ
.
TR θ T = R −1
θ .
G D n

R 3
R 2
R 3
R 2
m + n m
n
(1, 1) (2, 0)
(−1, 1) (−1, −1)
{ 1 , 2 } { 1 , 2 }
1 = α 1 1 + α 2 2
2 = β 1 1 + β 2 2 ,
α 1 α 2 β 1 β 2
( )
α1 α
U = 2
.
β 1 β 2
1 2 1 2 U −1
( ) ( )
U −1 1 1
= .
2 2

U U −1
U U −1 UU −1 = I
(UU −1 )=(U) (U −1 )=1;
(U) =±1 ±1
( ) 3 1
5 2
(−1, 1)
(1, 1)
(2, 0)
(−1, −1)
R 2
E(2) R 2
R 3
G ⊂ E(2) {(I,t):t ∈ L} L
R 2 Z × Z
G G 0 = {A :(A, b) ∈ G b} G 0
O(2) L G
H G 0 (A, ) G
(A, )(I,)(A, ) −1 =(A, A + )(A −1 , −A −1 )
=(AA −1 , −AA −1 + A + )
=(I,A);
(I,A) G A
L G 0 G
T G G/T ∼ = G 0

Z n D n
n =1, 2, 3, 4, 6
R 2
Z 1
Z 2
Z 3
Z 4
Z 6
D 1
D 1
D 1
D 2
D 2
D 2
D 2
D 3
D 4
D 6

p4m
p4g
n
R 3
R 4
R 5
R 3
〈, 〉 = 1 [
‖ + ‖ 2 −‖‖ 2 −‖‖ 2] .
2
O(n)
SO(n)
( √ √ )
1/ 2 −1/ 2
1/ √ 2 1/ √ 2
( √ √ )
1/ 5 2/ 5
−2/ √ 5 1/ √ 5

⎛
4/ √ 5 0 3/ √ ⎞
5
⎝−3/ √ 5 0 4/ √ 5⎠
0 −1 0
⎛
1/3 2/3
⎞
−2/3
⎝−2/3 2/3 1/3 ⎠
−2/3 1/3 2/3
R n α ∈ R
〈, 〉 = 〈, 〉
〈, + 〉 = 〈, 〉 + 〈, 〉
〈α, 〉 = 〈,α〉 = α〈, 〉
〈, 〉 ≥0 =0
〈, 〉 =0 R n =0
E(n) ={(A, ) :A ∈ O(n) ∈ R n }
{(2, 1), (1, 1)} {(12, 5), (7, 3)}
G E(2) T G
G G/T
A ∈ SL 2 (R)
R 2
A A
SO(n) O(n)
f R n
E(2) (A, ) ≠0
O(n)
=(x 1 ,x 2 ) R 2 x 2 1 + x2 2 =1A ∈ O(2)
A
G H
N G N H

H ∩ N = {}
HN = G
S 3 A 3 H = {(1), (12)}
Q 8
E(2) O(2) H H
R 2

13
n
Z n
G = H n ⊃ H n−1 ⊃···⊃H 1 ⊃ H 0 = {e},
H i H i+1 H i+1 /H i
G
Z p p Z mn
∼ = Zm × Z n
(m, n) =1
Z p
α 1
1
×···×Z p
αn
n ,
p k
G
{g i } G i I
G g i
G g i G G G
{g i : i ∈ I} g i G
{g i : i ∈ I} G G
S 3 (12) (123) Z × Z n
{(1, 0), (0, 1)}
Q
Q

p 1 /q 1 ,...,p n /q n p i /q i p
q 1 ,...,q n 1/p
Q p 1 /q 1 ,...,p n /q n p
p i /q i + p j /q j =(p i q j + p j q i )/(q i q j ).
H G {g i ∈ G : i ∈ I}
h ∈ H
h = g α 1
i 1
g ik
···g αn
i n
,
K g α 1
i 1 ···g αn
i n
g ik
K H K
G K = H H
g i
K gi 0 =1
K g = g k 1
i 1 ···g kn
i n
K
K
g −1 =(g k 1
i 1 ···g kn
i n
) −1 =(g −kn
i n
···g −k 1
i 1
).
g i
g i
a −3 b 5 a 7
a 4 b 5
p
G p G
p Z 2 × Z 2 Z 4 2 Z 27 3
p
G
Z p
α 1
1
× Z p
α 2
2
p i
×···×Z p
αn
n
540 = 2 2 ·3 3 ·5
Z 2 × Z 2 × Z 3 × Z 3 × Z 3 × Z 5
Z 2 × Z 2 × Z 3 × Z 9 × Z 5
Z 2 × Z 2 × Z 27 × Z 5
Z 4 × Z 3 × Z 3 × Z 3 × Z 5
Z 4 × Z 3 × Z 9 × Z 5

Z 4 × Z 27 × Z 5
G n p n
G p
n =1
G n k
k

G G i G
i =1,...,n p 0 i =1 1 ∈ G ig ∈ G i
p r i g−1 p r i h ∈ G i p s i
t r s
(gh) pt i = g
p t i h
p t i =1· 1=1,
G = G 1 G 2 ···G k
G i ∩ G j = {1} i ≠ j g 1 ∈ G 1
G 2 ,G 3 ,...,G k g 1 = g 2 g 3 ···g k g i ∈ G i g i p α i
g pα i
i
=1 i =2, 3,...,k g pα 2
2 ···pα k
k
1 =1 g 1 p 1
(p 1 ,p α 2
2 ···pα k
k )=1 g 1 =1 G 1
G 2 ,G 3 ,...,G k
G i ∩ G j = {1} i ≠ j
g ∈ G g 1 ···g k
g i ∈ G i g G
|g| = p β 1
1 pβ 2
2 ···pβ k
k
β 1 ,...,β k a i = |g|/p β i
i a i
b 1 ,...,b k a 1 b 1 + ···+ a k b k =1
g = g a 1b 1 +···+a k b k
= g a 1b1
···g a kb k
.
g (a ib i )p β i
i = g b i|g| = e,
g a ib i
G i g i = g a ib i
g = g 1 ···g k ∈ G 1 G 2 ···G k
G = G 1 G 2 ···G k
p i G i
G p g ∈ G
G 〈g〉×H H G
G p n
n n =1 G p g
k 1 ≤ k

a p a /∈ 〈g〉 h
〈g〉 |H| = p
gH G/H
g G |gH| < |g| = p m
H =(gH) pm−1 = g pm−1 H;
g pm−1 〈g〉 ∩H = {e} g
p m gH G/H
G/H ∼ = 〈gH〉×K/H
K G H 〈g〉∩K = {e} b ∈〈g〉∩K
bH ∈〈gH〉∩K/H = {H} b ∈〈g〉∩H = {e} G = 〈g〉K
G ∼ = 〈g〉×K
G g
G 〈g〉 = G G ∼ = Z |g| × H
H G |H| < |G|
G
Z p
α 1
1
× Z p
α 2
2
×···×Z p
αn
n
p i
× Z ×···×Z,
G
G = H n ⊃ H n−1 ⊃···⊃H 1 ⊃ H 0 = {e},
H i H i+1 H i G
Z ⊃ 9Z ⊃ 45Z ⊃ 180Z ⊃{0},
Z 24 ⊃〈2〉 ⊃〈6〉 ⊃〈12〉 ⊃{0}.
D 4
D 4 ⊃{(1), (12)(34), (13)(24), (14)(23)} ⊃{(1), (12)(34)} ⊃{(1)}.
{(1), (12)(34)} D 4

{K j }
{H i } {H i }⊂{K j } H i K j
Z ⊃ 3Z ⊃ 9Z ⊃ 45Z ⊃ 90Z ⊃ 180Z ⊃{0}
Z ⊃ 9Z ⊃ 45Z ⊃ 180Z ⊃{0}.
{H i } G
H i+1 /H i {H i }
{K j } G
{H i+1 /H i } {K j+1 /K j }
Z 60
Z 60 ⊃〈3〉 ⊃〈15〉 ⊃{0}
Z 60 ⊃〈4〉 ⊃〈20〉 ⊃{0}
Z 60 /〈3〉 ∼ = 〈20〉/{0} ∼ = Z 3
〈3〉/〈15〉 ∼ = 〈4〉/〈20〉 ∼ = Z 5
〈15〉/{0} ∼ = Z 60 /〈4〉 ∼ = Z 4 .
{H i } G
{H i } G
Z 60
Z 60 ⊃〈3〉 ⊃〈15〉 ⊃〈30〉 ⊃{0}
Z 60 /〈3〉 ∼ = Z 3
〈3〉/〈15〉 ∼ = Z 5
〈15〉/〈30〉 ∼ = Z 2
〈30〉/{0} ∼ = Z 2 .
Z 60
n ≥ 5
Z 60 ⊃〈2〉 ⊃〈4〉 ⊃〈20〉 ⊃{0}
S n ⊃ A n ⊃{(1)}
S n S n /A n
∼ = Z2 A n

{0} = H 0 ⊂ H 1 ⊂···⊂H n−1 ⊂ H n = Z
H 1 kZ
k ∈ N H 1 /H 0
∼ = kZ
Z 60
Z 60 Z 2 Z 2 Z 3 Z 5
G
G
k 1 ≤ k

H n−1 H n−1 K m−1 = G
K m−1 /(K m−1 ∩ H n−1 ) ∼ = (H n−1 K m−1 )/H n−1 = G/H n−1 .
G = H n ⊃ H n−1 ⊃ H n−1 ∩ K m−1 ⊃···⊃H 0 ∩ K m−1 = {e}
G = K m ⊃ K m−1 ⊃ K m−1 ∩ H n−1 ⊃···⊃K 0 ∩ H n−1 = {e}
G {H i }
H i+1 /H i
S 4
S 4 ⊃ A 4 ⊃{(1), (12)(34), (13)(24), (14)(23)} ⊃{(1)}
n ≥ 5
S n ⊃ A n ⊃{(1)}
S n S n
n ≥ 5
16
40
200
720
Z 12
S 3 × Z 4
Z 48
S 4
Q 8
S n n ≥ 5
D 4 Q
G = Z 2 × Z 2 ×···
G m n m G
n
G G

G H K G × H ∼ = G × K
H ∼ = K
G H G × H
G N G
N
N G N G/N
G
N G N G/N G
G G
G = P n ⊃ P n−1 ⊃···⊃P 1 ⊃ P 0 = {e}
P i P i+1 P i+1 /P i
G G
G N G G/N
D n n
G N G
N G/N
G p H K H
K K H
G n ≥ 2 G
G ′ G
G a −1 b −1 ab a, b ∈ G
G G (0) = G G (1) = G ′ G (i+1) =(G (i) ) ′
G (i+1) (G (i) ) ′
G
G (0) = G ⊃ G (1) ⊃ G (2) ⊃···
G G (n) = {e} n
G n ≥ 2 G
H K G
H ∗ K ∗ H K
H ∗ (H ∩ K ∗ ) H ∗ (H ∩ K)
K ∗ (H ∗ ∩ K) K ∗ (H ∩ K)
H ∗ (H ∩ K)/H ∗ (H ∩ K ∗ ) ∼ = K ∗ (H ∩ K)/K ∗ (H ∗ ∩ K) ∼ = (H ∩ K)/(H ∗ ∩ K)(H ∩ K ∗ )
G

n
n

14
G X
g ∈ G x ∈ X gx X
X G G X G × X → X
(g, x) ↦→ gx
ex = x x ∈ X
(g 1 g 2 )x = g 1 (g 2 x) x ∈ X g 1 ,g 2 ∈ G
X G X
G G X
(g, x) ↦→ x X
G
G = GL 2 (R) X = R 2 G X
v ∈ R 2 I Iv = v A B 2×2
(AB)v = A(Bv)
G = D 4 X = {1, 2, 3, 4}
D 4
{(1), (13), (24), (1432), (1234), (12)(34), (14)(23), (13)(24)}.
D 4 X (13)(24) 1
3 2 4
X G S X
X X G
σ ∈ G x ∈ X
(σ, x) ↦→ σ(x)

X = G G
(g, x) ↦→ λ g (x) =gx λ g
e · x = λ e x = ex = x
(gh) · x = λ gh x = λ g λ h x = λ g (hx) =g · (h · x).
H G G H H
G X = G H G
G H H G
H × G → G,
(h, g) ↦→ hgh −1
h ∈ H g ∈ G
(h 1 h 2 ,g)=h 1 h 2 g(h 1 h 2 ) −1
= h 1 (h 2 gh −1
2 )h−1 1
=(h 1 , (h 2 ,g)),
H G L H H L H
G
(g, xH) ↦→ gxH.
(gg ′ )xH = g(g ′ xH)
G X x, y ∈ X x G y
g ∈ G gx = y x ∼ G y x ∼ y G
X G G X
∼ ex = x x ∼ y x, y ∈ X
g gx = y g −1 y = x y ∼ x
x ∼ y y ∼ z
g h gx = y hy = z z = hy =(hg)x x z
X G X G
X G x X O x
G
G = {(1), (123), (132), (45), (123)(45), (132)(45)}
X = {1, 2, 3, 4, 5} X G O 1 = O 2 = O 3 = {1, 2, 3}
O 4 = O 5 = {4, 5}
G X g G
g X X g x ∈ X gx = x
g x ∈ X G
x x G x

X g ⊂ X G x ⊂ G
X = {1, 2, 3, 4, 5, 6} G
{(1), (12)(3456), (35)(46), (12)(3654)}.
X G
X (1) = X,
X (35)(46) = {1, 2},
X (12)(3456) = X (12)(3654) = ∅,
G 1 = G 2 = {(1), (35)(46)},
G 3 = G 4 = G 5 = G 6 = {(1)}.
G x G x ∈ X
G X x ∈ X
x G x G
e ∈ G x X g, h ∈ G x
gx = x hx = x (gh)x = g(hx) =gx = x
G x G x g ∈ G x x = ex =(g −1 g)x =(g −1 )gx = g −1 x g −1
G x
g ∈ G
|X g | x ∈ X |O x |
x ∈ X G x
G
G X G x ∈ X |O x | =[G :
G x ]
|G|/|G x | G x G
φ O x X
L Gx G x G y ∈O x g G
gx = y φ φ(y) =gG x φ φ(y 1 )=φ(y 2 )
φ(y 1 )=g 1 G x = g 2 G x = φ(y 2 ),
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
y 2 = g 2 x = g 1 gx = g 1 x = y 1 ;
φ φ
gG x gx = y φ(y) =gG x

X G X G X
X
X G = {x ∈ X : gx = x g ∈ G}.
|X| = |X G | +
n∑
|O xi |,
x k ,...,x n X
G (g, x) ↦→ gxg −1
G
Z(G) ={x : xg = gx g ∈ G},
G x 1 ,...,x k
G |O x1 | = n 1 ,...,|O xk | = n k
i=k
|G| = |Z(G)| + n 1 + ···+ n k .
x i C(x i )={g ∈ G : gx i = x i g}
x i
|G| = |Z(G)| +[G : C(x 1 )] + ···+[G : C(x k )].
G
S 3
6=1+2+3
{(1)}, {(123), (132)}, {(12), (13), (23)}.
D 4 {(1), (13)(24)}
{(13), (24)}, {(1432), (1234)}, {(12)(34), (14)(23)}.
D 4 8=2+2+2+2
S n
σ =(a 1 ,...,a k ) τ ∈ S n
τστ −1 =(τ(a 1 ),...,τ(a k )).
σ = σ 1 σ 2 ···σ r
σ i r i σ
τ ∈ S n
S n n
S 3
3
3=1+1+1
3=1+2

3=3;
n
n
G p n p G
|G| = |Z(G)| + n 1 + ···+ n k .
n i > 1 n i ||G| p n i p ||G| p
|Z(G)| G |Z(G)| ≥1
|Z(G)| ≥p g ∈ Z(G) g ≠1
G p 2 p G
|Z(G)| = p p 2 |Z(G)| = p 2
|Z(G)| = p Z(G) G/Z(G) p
aZ(G) G/Z(G) gZ(G)
a m Z(G) m g = a m x x
G hZ(G) ∈ G/Z(G) y Z(G) h = a n y
n x y G G
gh = a m xa n y = a m+n xy = a n ya m x = hg,
G
2 4 =16
90 ◦
B
W
W
B
W
W
W
W
W
W
W
W
B
W
W
B

X G x ∼ y G x G y
|G x | = |G y |
G X (g, x) ↦→ g · x x ∼ y g ∈ G
g · x = y a ∈ G x
gag −1 · y = ga · g −1 y = ga · x = g · x = y,
φ : G x → G y φ(a) =gag −1 φ
φ(ab) =gabg −1 = gag −1 gbg −1 = φ(a)φ(b).
φ(a) =φ(b) gag −1 = gbg −1 a = b
φ b G y g −1 bg G x
φ(g −1 bg) =b
g −1 bg · x = g −1 b · gx = g −1 b · y = g −1 · y = x;
G X k
X
k = 1 ∑
|X g |.
|G|
g∈G
x g ∈ G
g x gx = x
g x
∑
|X g |.
∑
|G x |;
g∈G
x∈X
∑ g∈G |X g| = ∑ x∈X |G x|
∑
|G y | = |O x |·|G x |.
y∈O x
|G|
k
∑
|X g | = ∑ |G x | = k ·|G|.
g∈G x∈X
X = {1, 2, 3, 4, 5} G
G = {(1), (13), (13)(25), (25)} X {1, 3} {2, 5} {4}
X (1) = X

k = 1
|G|
X (13) = {2, 4, 5}
X (13)(25) = {4}
X (25) = {1, 3, 4}.
∑
|X g | = 1 4 (5+3+1+3)=3.
g∈G
D 4
(1) (13) (24) (1432)
(1234) (12)(34) (14)(23) (13)(24)
G {1, 2, 3, 4}
X Y = {B,W} B W
f : X → Y
σ ∈ D 4 ˜σ
˜σ(f) =f ◦ σ f : X → Y f
f(1) = B
f(2) = W
f(3) = W
f(4) = W
σ = (12)(34) ˜σ(f) =f ◦ σ 2 B
W ˜σ ˜G ˜X
˜X X
Y ˜G
˜X(1) = ˜X | ˜X| =2 4 = 16
˜X(1234) f ∈ ˜X f (1234)
f(1) = f(2) = f(3) = f(4) f
f(x) =B f(x) =W x | ˜X (1234) | =2
| ˜X (1432) | =2
˜X (13)(24) f(1) = f(3) f(2) = f(4) | ˜X (13)(24) | =2 2 =4
| ˜X (12)(34) | =4
| ˜X (14)(23) | =4

˜X (13) f(1) = f(3) | ˜X (13) | =
2 3 =8
| ˜X (24) | =8
1
8 (24 +2 1 +2 2 +2 1 +2 2 +2 2 +2 3 +2 3 )=6
G X ˜X
X Y ˜G ˜X ˜σ ∈ ˜G
˜σ(f) =f ◦ σ σ ∈ G f ∈ ˜X n
σ | ˜X σ | = |Y | n
σ ∈ G f ∈ ˜X f ◦ σ ˜X g
X Y ˜σ(f) =˜σ(g) x ∈ X
f(σ(x)) = ˜σ(f)(x) =˜σ(g)(x) =g(σ(x)).
σ X x ′ X x X σ
f g X f = g ˜σ
σ ↦→ ˜σ
σ X σ = σ 1 σ 2 ···σ n
f ˜X σ σ n |Y |
| ˜X σ | = |Y | n
X = {1, 2,...,7} Y = {A, B, C} g
X (13)(245) = (13)(245)(6)(7) n =4 f ∈ ˜X g
g |Y | =3
| ˜X g | =3 4 =81
1
8 (44 +4 1 +4 2 +4 1 +4 2 +4 2 +4 3 +4 3 )=55
n

x 1
x 2
f f(x 1 ,x 2 ,...,x n )
x n
n
n Z n 2
Z 2 n
2 n n 2 2n n
a
b
f f(a, b)
a
b
f f(b, a) =g(a, b)
a b
f g g f
g(a, b, c) =f(b, c, a)
g ∼ f (acb)
(ab)
f 2 ∼ f 4
f 3 ∼ f 5
f 10 ∼ f 12
f 11 ∼ f 13 .
f 0 f 1 f 2 f 3 f 4 f 5 f 6 f 7
f 8 f 9 f 10 f 11 f 12 f 13 f 14 f 15
2 23 = 256
2 24 =65,536

a b c
f g
f g
{a, b, c}
a b c
2 3
(a, b, c) (acb)
(0, 0, 0) ↦→ (0, 0, 0)
(0, 0, 1) ↦→ (0, 1, 0)
(0, 1, 0) ↦→ (1, 0, 0)
(1, 1, 0) ↦→ (1, 0, 1)
(1, 1, 1) ↦→ (1, 1, 1).
X n |X| =2 n
(0,...,0, 1) ↦→ 0
(0,...,1, 0) ↦→ 1
(0,...,1, 1) ↦→ 2
(1,...,1, 1) ↦→ 2 n − 1.
(a), (ac), (bd), (adcb),
(abcd), (ab)(cd), (ad)(bc), (ac)(bd).
1
8 (216 +2· 2 12 +2· 2 6 +3· 2 10 ) = 9616

(a) (0)
(ac) (2, 8)(3, 9)(6, 12)(7, 13)
(bd) (1, 4)(3, 6)(9, 12)(11, 14)
(adcb) (1, 2, 4, 8)(3, 6.12, 9)(5, 10)(7, 14, 13, 11)
(abcd) (1, 8, 4, 2)(3, 9, 12, 6)(5, 10)(7, 11, 13, 14)
(ab)(cd) (1, 2)(4, 8)(5, 10)(6, 9)(7, 11)(13, 14)
(ad)(bc) (1, 8)(2, 4)(3, 12)(5, 10)(7, 14)(11, 13)
(ac)(bd) (1, 4)(2, 8)(3, 12)(6, 9)(7, 13)(11, 14)
G
G = H n ⊃ H n−1 ⊃···⊃H 1 ⊃ H 0 = {e}
H i H i+1 H i+1 /H i
G
X G
G
X g G x
X = {1, 2, 3} G = S 3 = {(1), (12), (13), (23), (123), (132)}
X = {1, 2, 3, 4, 5, 6} G = {(1), (12), (345), (354), (12)(345), (12)(354)}
G X G
x ∈ X |G| = |O x |·|G x |

G θ ∈ G
R 2 θ
P
R 2 G
P
G P
G = A 4 G (g, h) ↦→ ghg −1
G
G
S 4 D 5 Z 9 Q 8
S 5 A 5
1,...,6
12
CH 3

H
H
H
H
H
H
x 1 x 2 x 3 S 3 x 1
x 2 x 3 x 4 S 4
x 1 x 2 x 3 x 4 S 4
(x 1 x 2 x 3 x 4 )
12
G X G
X G X
G X
p p n
S n
a ∈ G g ∈ G gC(a)g −1 = C(gag −1 )
|G| = p n p |Z(G)|

15
G
m n m G n
A 4 12 6
G
G (g, x) ↦→ gxg −1 x 1 ,...,x k
G
|G| = |Z(G)| +[G : C(x 1 )] + ···+[G : C(x k )],
Z(G) ={g ∈ G : gx = xg x ∈ G} G C(x i )={g ∈ G :
gx i = x i g} x i
p
p G p G
p p G p p
G p p
G G p
G |G| = p G
k p ≤ k

G Z(G) Z(G) p
G
p
G G p |G| = p n
A 5 |A 5 | =60=2 2 · 3 · 5
A 5 2 3 5
A 5
G p
p r |G| G p r
G |G| = p
G n n>p
n p n
|G| = |Z(G)| +[G : C(x 1 )] + ···+[G : C(x k )].
p [G : C(x i )] i p r ||C(x i )| p r
|G| = |C(x i )|·[G : C(x i )] C(x i )
p [G : C(x i )] i p |G|
p |Z(G)| Z(G)
p g N g N
Z(G) Z(G) N G Z(G)
G G/N |G|/p
G/N H p r−1
H φ : G → G/N p r G
p P G p G
G S G
H S H H S
H ×S →S
h · K ↦→ hKh −1
K S
N(H) ={g ∈ G : gHg −1 = H}
G H G H
N(H) N(H) G H
P p G x
p x −1 Px = P x ∈ P
x ∈ N(P ) 〈xP 〉⊂N(P )/P
p H N(P )
P H/P = 〈xP 〉 |H| = |P |·|〈xP 〉| H
p P p H P
p |G| H = P H/P xP = P
x ∈ P

H K G H
K [H : N(K) ∩ H]
K
N(K) ∩ H h −1 Kh ↦→ (N(K) ∩ H)h h 1 ,h 2 ∈ H
(N(K) ∩ H)h 1 =(N(K) ∩ H)h 2 h 2 h −1
1 ∈ N(K)
K = h 2 h −1
1 Kh 1h −1
2 h −1
1 Kh 1 = h −1
2 Kh 2
H K N(K) ∩ H H
G p
|G| p G P 1 P 2
p g ∈ G gP 1 g −1 = P 2
P p G |G| = p r m |P | = p r
S = {P = P 1 ,P 2 ,...,P k }
P G k =[G : N(P )]
|G| = p r m = |N(P )|·[G : N(P )] = |N(P )|·k.
p r |N(P )| p k
p Q Q ∈S Q
P i S
P i [Q : N(P i ) ∩ Q]
|Q| =[Q : N(P i ) ∩ Q]|N(P i ) ∩ Q| [Q : N(P i ) ∩ Q] |Q| = p r
p p k
p P j x −1 P j x = P j x ∈ Q
P j = Q
G p
G p 1( p)
|G|
P p p
S = {P = P 1 ,P 2 ,...,P k },
P P
P p P
p |S| {P }
|S| p 1 |S| ≡ 1( p)
G S p
P ∈S
|S| = | P | =[G : N(P )]
[G : N(P )] |G|
p

A 5
2 3 4 5 p A 5 3 4 5
p A 5
5 60 1( 5)
5 A 5 5
5
A 5 A 5
5 A 5
p q p

5 · 7 · 47 = 1645
G ′ = 〈aba −1 b −1 : a, b ∈ G〉
aba −1 b −1 G G ′
G G/G ′
G ′ G G
5 · 7 · 47 = 1645
G H 1
47 G/H 1
G H |G ′ | 1 47
|G ′ | =1 |G ′ | =47
G 5 7
H 2 H 3 G |H 2 | =5 |H 3 | =7
G ′ H i i =1, 2 G ′ 1
5 7 |G ′ | =1 47
G G
A 5
G 20
G 5
1( 5) 20 1
5 5
G p n n>1 p
G G G
4 8 9 16 25 27 32 49 64 81
4 9 25 49
56 = 2 3 · 7
p p
7 7
7
8 · 6=48
7 2
2 2
2 48 7
7 2 8
2 56 2

G G G
48
48
H K G
|HK| = |H|·|K|
|H ∩ K| .
HK = {hk : h ∈ H, k ∈ K}.
|HK|≤|H|·|K| HK
H K h 1 k 1 = h 2 k 2 h 1 ,h 2 ∈ H
k 1 ,k 2 ∈ K
a =(h 1 ) −1 h 2 = k 1 (k 2 ) −1 .
a ∈ H ∩ K (h 1 ) −1 h 2 H k 2 (k 1 ) −1 K
h 2 = h 1 a −1
k 2 = ak 1 .
h = h 1 b −1 k = bk 1 b ∈ H ∩ K hk = h 1 k 1 h ∈ H
k ∈ K hk ∈ HK h i k i h i ∈ H
k i ∈ K H ∩ K |H ∩ K|
|HK| =(|H|·|K|)/|H ∩ K|
G 48
G 8 16
G 2 16
2 H
K |H ∩ K| =8|H ∩ K| ≤4
16 · 16
|HK| = =64,
4
H ∩ K H K
H K H ∩ K H
N(H ∩ K) N(H ∩ K) 16 |N(H ∩ K)|
16 1 48
|N(H ∩ K)| =48 N(H ∩ K) =G
p p
p

p G 18 24 54 72
80
3 S 4
45 9
H p G H p G
N(H)
96
160
H G |H| = p k p
H p G
G p 2 q 2 p q q ∤ p 2 − 1
p ∤ q 2 − 1 G
33 3
H G H
G
G p
p G G
G p r p G
p r−1
G p n k kq G
H G
[G : N(H)]
2 S 5 D 4
p p m
( p k )
m
p ∤
p k .

S p k G p |S|
G S aT = {at : t ∈ T } a ∈ G
T ∈S
p ∤ |O T | T ∈S
{T 1 ,...,T u } p ∤ u H = {g ∈ G : gT 1 = T 1 }
H G |G| = u|H|
p k |H| p k ≤|H|
|H| = |O T |≤p k p k = |H|
G G ′ = 〈aba −1 b −1 : a, b ∈ G〉 G
G/G ′ {aba −1 b −1 : a, b ∈ G}
60
1 16 14 31 1 46 2
2 17 1 32 51 47 1
3 18 33 1 48 52
4 19 34 49
5 20 5 35 1 50 5
6 21 36 14 51
7 22 2 37 1 52
8 23 1 38 53
9 24 39 2 54 15
10 25 2 40 14 55 2
11 26 2 41 1 56
12 5 27 5 42 57 2
13 28 43 1 58
14 29 1 44 4 59 1
15 1 30 4 45 60 13
G |G| ≤60
G |G| ≤60
G G n n =1,...,60
n

16
R
a + b = b + a a, b ∈ R
(a + b)+c = a +(b + c) a, b, c ∈ R
0 R a +0=a a ∈ R
a ∈ R −a R a +(−a) =0
(ab)c = a(bc) a, b, c ∈ R
a, b, c ∈ R
a(b + c) =ab + ac
(a + b)c = ac + bc.
(R, +)
1 ∈ R 1 ≠0 1a = a1 =a a ∈ R
R R ab = ba a, b R
R
a, b ∈ R ab =0 a =0 b =0
R R
a ∈ R a ≠0 a −1 a −1 a = aa −1 =1

Z
ab =0 a b a =0 b =0
Z 2
1/2 1 −1
Q R
C
a b Z n ab ( n)
Z 12 5 · 7 ≡ 11 ( 12) Z n
Z n
3 · 4 ≡ 0( 12) Z 12
a R
b R ab =0 3 4 Z 12
[a, b]
f(x) =x 2 g(x) = x (f + g)(x) =f(x)+g(x) =x 2 + x
(fg)(x) =f(x)g(x) =x 2 x
2×2 R
AB ≠ BA AB =0 A B
( ) ( ) ( ) ( )
1 0
0 1 0 i
i 0
1= , = , = , = ,
0 1 −1 0 i 0
0 −i
i 2 = −1
2 = 2 = 2 = −1
=
=
=
= −

= −
= −.
H a + b + c + d a, b, c, d
H 2 × 2
( ) α β
,
−β α
α = a + di β = b + ci
H 1
(a 1 + b 1 + c 1 + d 1 )+(a 2 + b 2 + c 2 + d 2 )
=(a 1 + a 2 )+(b 1 + b 2 ) +(c 1 + c 2 ) +(d 1 + d 2 )
(a 1 + b 1 + c 1 + d 1 )(a 2 + b 2 + c 2 + d 2 )=α + β + γ + δ,
α = a 1 a 2 − b 1 b 2 − c 1 c 2 − d 1 d 2
β = a 1 b 2 + a 2 b 1 + c 1 d 2 − d 1 c 2
γ = a 1 c 2 − b 1 d 2 + c 1 a 2 + d 1 b 2
δ = a 1 d 2 + b 1 c 2 − c 1 b 2 + d 1 a 2 .
H
H
(a + b + c + d)(a − b − c − d) =a 2 + b 2 + c 2 + d 2 .
a b c d a + b + c + d ≠0
( )
a − b − c − d
(a + b + c + d)
a 2 + b 2 + c 2 + d 2 =1.
R a, b ∈ R
a0 =0a =0
a(−b) =(−a)b = −ab
(−a)(−b) =ab
a0 =a(0 + 0) = a0+a0;
a0 =0 0a =0 ab + a(−b) =a(b − b) =a0 =0
−ab = a(−b) −ab =(−a)b
(−a)(−b) =−(a(−b)) = −(−ab) =ab

S R S R S
R
nZ Z
Z ⊂ Q ⊂ R ⊂ C.
R S R S R
S ≠ ∅
rs ∈ S r, s ∈ S
r − s ∈ S r, s ∈ S
R = M 2 (R) 2 × 2 R T
R
{( )
}
a b
T = : a, b, c ∈ R ,
0 c
T R
( )
( a b
a
′
b
A = B =
′ )
0 c
0 c ′
T A − B T
( aa
′
ab
AB =
′ + bc ′ )
0 cc ′
T
R r R
r s ∈ R rs =0
a R a
R R
i 2 = −1 Z[i] ={m + ni : m, n ∈ Z}
α = a + bi
Z[i] α = a − bi αβ =1 αβ =1β = c + di
1=αβαβ =(a 2 + b 2 )(c 2 + d 2 ).
a 2 + b 2 1 −1 a + bi = ±1 a + bi = ±i
±1 ±i

{( ) 1 0
F = ,
0 1
Z 2
( ) 1 1
,
1 0
( ) 0 1
,
1 1
( )} 0 0
0 0
Q( √ 2) = {a + b √ 2:a, b ∈ Q}
a + b √ 2 Q( √ 2)
a
a 2 − 2b 2 + −b √
2.
a 2 − 2b 2
D
D a ∈ D ab = ac
b = c
D D ab = ac
a ≠0 a(b − c) =0 b − c =0 b = c
D
ab = ac b = c ab =0a ≠0 ab = a0 b =0 a
D D ∗ D
D ∗ a ∈ D ∗
λ a : D ∗ → D ∗ λ a (d) =ad a ≠0 d ≠0
ad ≠0 λ a d 1 ,d 2 ∈ D ∗
ad 1 = λ a (d 1 )=λ a (d 2 )=ad 2
d 1 = d 2 D ∗ λ a
d ∈ D ∗ λ a (d) =ad =1 a D
d a D
n r R r + ···+ r n
nr R n
nr =0 r ∈ R R
0 R R
p Z p p
Z p Z p a
pa =0 Z p
p
R 1 n
R n
1 n n n1 =0
r ∈ R
nr = n(1r) =(n1)r =0r =0.
n n1 =0 R

D D n
n ≠0 n n = ab 1

φ : R → S
R φ(R)
φ(0) = 0
1 R 1 S R S φ φ(1 R )=1 S
R φ(R) ≠ {0} φ(R)
R I R a I r R
ar ra I rI ⊂ I Ir ⊂ I r ∈ R
R {0} R
R I R 1 I
r ∈ R r1 =r ∈ I I = R
a R
〈a〉 = {ar : r ∈ R}
R 〈a〉 0=a0 a = a1 〈a〉
〈a〉 〈a〉 ar + ar ′ = a(r + r ′ ) ar
−ar = a(−r) ∈〈a〉 ar ∈〈a〉
s ∈ R s(ar) =a(sr) 〈a〉
R 〈a〉 = {ar : r ∈ R}
Z
{0} 〈0〉 = {0} I
Z I m n
I a I
q r
a = nq + r
0 ≤ r

I ⊂ I Ir ⊂ I
r ∈ R
rI ⊂ I Ir ⊂ I r ∈ R
I R R/I
(r + I)(s + I) =rs + I.
R/I r + I
s + I R/I (r + I)(s + I) =rs + I
r ′ ∈ r + I s ′ ∈ s + I r ′ s ′ rs + I
r ′ ∈ r + I a I r ′ = r + a b ∈ I
s ′ = s + b
r ′ s ′ =(r + a)(s + b) =rs + as + rb + ab
as + rb + ab ∈ I I r ′ s ′ ∈ rs + I
R/I
I R φ : R → R/I φ(r) =r + I
R R/I I
φ : R → R/I
φ r s R
φ(r)φ(s) =(r + I)(s + I) =rs + I = φ(rs),
φ : R → R/I
ψ : R → S
ψ R φ : R → R/ ψ
η : R/ ψ → ψ(R) ψ = ηφ
K = ψ
η : R/K → ψ(R) η(r + K) =ψ(r)
R R/K
η((r + K)(s + K)) = η(r + K)η(s + K)
η((r + K)(s + K)) = η(rs + K)
= ψ(rs)
= ψ(r)ψ(s)
= η(r + K)η(s + K).

I R
J R I ∩ J I
I/I ∩ J ∼ = (I + J)/J.
R I J
R J ⊂ I
R/I ∼ = R/J
I/J .
I R
S ↦→ S/I S I
R/I R I
R/I
I R R/I
M R R M
R R M I
M I = R
R M R
M R R/M
M R R R/M
1+M R/M
R/M a + M R/M
a /∈ M I {ra + m : r ∈ R m ∈ M} I
R I 0a +0=0 I r 1 a + m 1 r 2 a + m 2
I
(r 1 a + m 1 ) − (r 2 a + m 2 )=(r 1 − r 2 )a +(m 1 − m 2 )
I r ∈ R rI ⊂ I I
I M M
I = R I m M b
R 1=ab + m
1+M = ab + M = ba + M =(a + M)(b + M).
M R/M R/M
0+M = M 1+M M
R I M I = R
a I M a + M
b + M R/M (a + M)(b + M) =ab + M =1+M
m ∈ M ab + m =1 1 I r1 =r ∈ I r ∈ R
I = R

pZ Z p pZ
Z/pZ ∼ = Z p
P R ab ∈ P
a ∈ P b ∈ P
P = {0, 2, 4, 6, 8, 10} Z 12
R 1 1 ≠0 P
R R/P
P R R/P
ab ∈ P a+P b+P R/P (a+P )(b+P )=0+P = P
a + P = P b + P = P a P b P
P
P
(a + P )(b + P )=ab + P =0+P = P.
ab ∈ P a /∈ P b P
b + P =0+P R/P
Z nZ Z/nZ ∼ = Z n
n
Z pZ p

m n (m, n) =1 a, b ∈ Z
x ≡ a ( m)
x ≡ b ( n)
x 1 x 2 x 1 ≡ x 2 ( mn)
x ≡ a ( m) a + km
k ∈ Z k 1
a + k 1 m ≡ b
( n).
k 1 m ≡ (b − a) ( n)
k 1 m n s t
ms + nt =1
(b − a)ms =(b − a) − (b − a)nt,
[(b − a)s]m ≡ (b − a) ( n).
k 1 =(b − a)s
mn c 1 c 2
c i ≡ a ( m)

i =1, 2
c i ≡ b ( n)
c 2 ≡ c 1 ( m)
c 2 ≡ c 1 ( n).
m n c 1 − c 2 c 2 ≡ c 1 ( mn)
x ≡ 3 ( 4)
x ≡ 4 ( 5).
s t 4s +5t =1
s =4 t = −3
x = a + k 1 m =3+4k 1 =3+4[(5− 4)4] = 19.
n 1 ,n 2 ,...,n k
(n i ,n j )=1 i ≠ j a 1 ,...,a k
x ≡ a 1 ( n 1 )
x ≡ a 2 ( n 2 )
x ≡ a k ( n k )
n 1 n 2 ···n k
k =2
k
x ≡ a 1 ( n 1 )
x ≡ a 2 ( n 2 )
x ≡ a k+1 ( n k+1 ).
k n 1 ···n k
a n 1 ···n k n k+1
x ≡ a ( n 1 ···n k )
x ≡ a k+1 ( n k+1 )
n 1 ...n k+1
x ≡ 3 ( 4)
x ≡ 4 ( 5)
x ≡ 1 ( 9)

x ≡ 5 ( 7).
19
19 ( 20)
x ≡ 19 ( 20)
x ≡ 1 ( 9)
x ≡ 5 ( 7).
x ≡ 19 ( 180)
x ≡ 5 ( 7).
19
1260
2 63 − 1=9,223,372,036,854,775,807.
2 511 − 1
2134 1531
95 97 98 99
2134 ≡ 44 ( 95)
2134 ≡ 0 ( 97)
2134 ≡ 76 ( 98)
2134 ≡ 55 ( 99)

1531 ≡ 11 ( 95)
1531 ≡ 76 ( 97)
1531 ≡ 61 ( 98)
1531 ≡ 46 ( 99).
2134 · 1531 ≡ 44 · 11 ≡ 9 ( 95)
2134 · 1531 ≡ 0 · 76 ≡ 0 ( 97)
2134 · 1531 ≡ 76 · 61 ≡ 30 ( 98)
2134 · 1531 ≡ 55 · 46 ≡ 55 ( 99).
2134·1531
x ≡ 9 ( 95)
x ≡ 0 ( 97)
x ≡ 30 ( 98)
x ≡ 55 ( 99).
95 · 97 ·
98 · 99 = 89,403,930 x 2134 · 1531 =
3,267,154
7Z
Z 18
Q( √ 2)={a + b √ 2:a, b ∈ Q}
Q( √ 2, √ 3)={a + b √ 2+c √ 3+d √ 6:a, b, c, d ∈ Q}
Z[ √ 3]={a + b √ 3:a, b ∈ Z}
R = {a + b 3√ 3:a, b ∈ Q}
Z[i] ={a + bi : a, b ∈ Z i 2 = −1}

Q( 3√ 3)={a + b 3√ 3+c 3√ 9:a, b, c ∈ Q}
R 2 × 2
( ) a b
,
0 0
a, b ∈ R R
S R
Z 10
Z 12
Z 7
M 2 (Z) 2 × 2 Z
M 2 (Z 2 ) 2 × 2 Z 2
Z 18
Z 25
M 2 (R) 2 × 2 R
M 2 (Z) 2 × 2 Z
Q
R I
R/I
R = Z I =6Z
R = Z 12 I = {0, 3, 6, 9}
φ : Z/6Z → Z/15Z
R C
Q( √ 2) = {a + b √ 2:a, b ∈ Q}
Q( √ 3)={a + b √ 3:a, b ∈ Q}
{( ) ( ) ( ) ( )}
1 0 1 1 0 1 0 0
F = , , ,
0 1 1 0 1 1 0 0
Z 2
φ : C → M 2 (R)
( ) a b
φ(a + bi) = .
−b a
φ C M 2 (R)
Z[i]
Z[ √ 3 i] ={a + b √ 3 i : a, b ∈ Z}

x ≡ 2 ( 5)
x ≡ 6 ( 11)
x ≡ 4 ( 7)
x ≡ 7 ( 9)
x ≡ 5 ( 11)
x ≡ 3 ( 7)
x ≡ 0 ( 8)
x ≡ 5 ( 15)
x ≡ 2 ( 4)
x ≡ 3 ( 5)
x ≡ 0 ( 8)
x ≡ 1 ( 11)
x ≡ 5 ( 13)
2234 + 4121
95 97 98 99
2134 ·
1531 98
99
R R {0} R
a R (−1)a = −a
φ : R → S
R φ(R)
φ(0) = 0
1 R 1 S R S φ φ(1 R )=1 S
R φ(R) ≠0 φ(R)
R/I
I R
J R I ∩ J I
I/I ∩ J ∼ = I + J/J.
R I J
R J ⊂ I
R/I ∼ = R/J
I/J .
I R S → S/I
S I
R/I R R/I
R S R S R
S ≠ ∅
rs ∈ S r, s ∈ S

− s ∈ S r, s ∈ S
R {R α } ⋂ R α
R
{I α } α∈A R ⋂ α∈A I α
R I 1 I 2 R I 1 ∪ I 2
R R {0} R
R
R a R a n =0
n R
R a ∈ R a 2 = a
R a 3 = a a ∈ R R
R 1 R S R 1 S
1 R =1 S
R 1=0
R = {0}
S R R ′ R
S
R R
Z(R) ={a ∈ R : ar = ra r ∈ R}.
Z(R) R
p
Z (p) = {a/b : a, b ∈ Z (b, p) =1}
Z (p) p
Z p
R
u R i u : R → R r ↦→ uru −1 i u
R R
R R (R)
R (R) (R)
(R)
U(R) R
φ : U(R) → (R)
u ↦→ i u φ
(Z) (Z) U(Z)
R S
R × S

(r, s)+(r ′ ,s ′ )=(r + r ′ ,s+ s ′ )
(r, s)(r ′ ,s ′ )=(rr ′ ,ss ′ )
x x 2 = x
0 1 x
(a, n) =d (b, d) ≠1 ax ≡ b ( n)
R I J
R I + J = R
r s R
x ≡ r ( I)
x ≡ s ( J)
I ∩ J
I J R I + J = R
R/(I ∩ J) ∼ = R/I × R/J.

17
p(x) =x 3 − 3x +2
q(x) =3x 2 − 6x +5,
p(x) +q(x) p(x)q(x)
(p + q)(x) =p(x)+q(x)
=(x 3 − 3x +2)+(3x 2 − 6x +5)
= x 3 +3x 2 − 9x +7
(pq)(x) =p(x)q(x)
=(x 3 − 3x + 2)(3x 2 − 6x +5)
=3x 5 − 6x 4 − 4x 3 +24x 2 − 27x +10.
R
f(x) =
n∑
a i x i = a 0 + a 1 x + a 2 x 2 + ···+ a n x n ,
i=0
a i ∈ R a n ≠0 R x
a 0 ,a 1 ,...,a n f a n
n
a n ≠0 f n
f(x) =n n f =0

f −∞
R R[x]
p(x) =a 0 + a 1 x + ···+ a n x n
q(x) =b 0 + b 1 x + ···+ b m x m ,
p(x) =q(x) a i = b i i ≥ 0
p(x) q(x)
p(x) =a 0 + a 1 x + ···+ a n x n
q(x) =b 0 + b 1 x + ···+ b m x m .
p(x)+q(x) =c 0 + c 1 x + ···+ c k x k ,
c i = a i + b i i p(x) q(x)
c i =
p(x)q(x) =c 0 + c 1 x + ···+ c m+n x m+n ,
i∑
a k b i−k = a 0 b i + a 1 b i−1 + ···+ a i−1 b 1 + a i b 0
k=0
i
p(x) =3+0x +0x 2 +2x 3 +0x 4
q(x) =2+0x − x 2 +0x 3 +4x 4
Z[x]
p(x) = 3 + 2x 3 q(x) =
2 − x 2 +4x 4
p(x)+q(x) =5− x 2 +2x 3 +4x 4 .
p(x)q(x) =(3+2x 3 )(2 − x 2 +4x 4 )=6− 3x 2 +4x 3 +12x 4 − 2x 5 +8x 7 ,
c i
p(x) =3+3x 3 q(x) =4+4x 2 +4x 4
Z 12 [x] p(x) q(x) 7+4x 2 +3x 3 +4x 4
R[x] R

R R[x]
R[x]
∑
f(x) =0 p(x) =
n
i=0 a ix i p(x) −p(x) = ∑ n
i=0 (−a i)x i = − ∑ n
i=0 a ix i
R
p(x) =
q(x) =
r(x) =
m∑
a i x i ,
i=0
n∑
b i x i ,
i=0
p∑
c i x i .
i=0
[( m
)(
∑
n∑
)] ( p∑
)
[p(x)q(x)]r(x) = a i x i b i x i c i x i
i=0 i=0
i=0
⎡ ⎛ ⎞ ⎤ )
m+n
∑ i∑
= ⎣ ⎝ a j b i−j
⎠ x i ⎦ c i x i
=
=
=
i=0
m+n+p
∑
i=0
m+n+p
∑
i=0
m+n+p
∑
i=0
j=0
j+k+l=i
( p∑
i=0
⎡ (
i∑ j∑
)
⎣ a k b j−k
j=0 k=0
⎛
⎞
⎝
∑
a j b k c l
⎠ x i
⎡
⎣
i∑
j=0
( m
) ⎡ ∑
= a i x i ⎣
=
i=0
( m
∑
c i−j
⎤
⎦ x i
( i−j
) ⎤ ∑
a j b k c i−j−k
⎦ x i
k=0
⎛ ⎞ ⎤
n+p
∑ i∑
⎝ b j c i−j
⎠ x i ⎦
i=0
)[( n∑
a i x i
i=0 i=0
= p(x)[q(x)r(x)]
j=0
b i x i )( p∑
i=0
c i x i )]
p(x) q(x) R[x] R
p(x)+ q(x) =(p(x)q(x)) R[x]
p(x) =a m x m + ···+ a 1 x + a 0

q(x) =b n x n + ···+ b 1 x + b 0
a m ≠0 b n ≠0 p(x) q(x) m n
p(x)q(x) a m b n x m+n R
p(x)q(x) m + n p(x)q(x) ≠0 p(x) ≠0 q(x) ≠0
p(x)q(x) ≠0 R[x]
x 2 − 3xy +2y 3
R x y
(R[x])[y]
(R[x])[y] ∼ = R([y])[x]
R[x, y] R[x, y] x
y R n
R R[x 1 ,x 2 ,...,x n ]
R α ∈ R
φ α : R[x] → R
p(x) =a n x n + ···+ a 1 x + a 0
φ α (p(x)) = p(α) =a n α n + ···+ a 1 α + a 0 ,
p(x) = ∑ n
i=0 a ix i q(x) = ∑ m
i=0 b ix i φ α (p(x) +
q(x)) = φ α (p(x)) + φ α (q(x)) φ α
φ α (p(x))φ α (q(x)) = p(α)q(α)
( n∑
)( m
)
∑
= a i α i b i α i
i=0
i=0
( i∑
)
a k b i−k
k=0
=
m+n
∑
i=0
= φ α (p(x)q(x)).
φ α : R[x] → R α
α i
a b
b>0 q r a = bq + r 0 ≤ r

q(x) r(x) f(x)
0=0· g(x)+0;
q r f(x)
f(x) =n g(x) =m m>n
q(x) =0 r(x) =f(x) m ≤ n
n
f(x) =a n x n + a n−1 x n−1 + ···+ a 1 x + a 0
g(x) =b m x m + b m−1 x m−1 + ···+ b 1 x + b 0
f ′ (x) =f(x) − a n
x n−m g(x)
b m
n q ′ (x)
r(x)
f ′ (x) =q ′ (x)g(x)+r(x),
r(x) =0 r(x) g(x)
q(x) =q ′ (x)+ a n
x n−m .
b m
f(x) =g(x)q(x)+r(x),
r(x) r(x) < g(x)
q(x) r(x)
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
f(x) =g(x)q(x)+r(x) =g(x)q 1 (x)+r 1 (x),
g(x)[q(x) − q 1 (x)] = r 1 (x) − r(x).
q(x) − q 1 (x)
(g(x)[q(x) − q 1 (x)]) = (r 1 (x) − r(x)) ≥ g(x).
r(x) r 1 (x) g(x)
r(x) =r 1 (x) q(x) =q 1 (x)
x 3 − x 2 +2x − 3 x − 2
x 2 + x + 4
x − 2 x 3 − x 2 + 2x − 3
x 3 − 2x 2
x 2 + 2x − 3
x 2 − 2x
4x − 3
4x − 8
5
x 3 − x 2 +2x − 3=(x − 2)(x 2 + x +4)+5

p(x) F [x] α ∈ F α p(x)
p(x) φ α
α p(x) p(α) =0
F α ∈ F p(x) ∈ F [x]
x − α p(x) F [x]
α ∈ F p(α) = 0
q(x) r(x)
p(x) =(x − α)q(x)+r(x)
r(x) x − α r(x)
r(x) =a a ∈ F
p(x) =(x − α)q(x)+a.
0=p(α) =0· q(α)+a = a;
p(x) =(x − α)q(x) x − α p(x)
x − α p(x) p(x) = (x − α)q(x)
p(α) =0· q(α) =0
F p(x) n F [x]
n F
p(x) p(x) =0 p(x)
p(x) =1 p(x) =ax + b a
b F α 1 α 2 p(x) aα 1 + b = aα 2 + b α 1 = α 2
p(x) > 1 p(x) F
α p(x) p(x) =(x − α)q(x) q(x) ∈ F [x]
q(x) n − 1 β
p(x) α p(β) =(β − α)q(β) =0 α ≠ β F
q(β) =0 q(x) n − 1 F
α p(x) n F
F d(x)
p(x),q(x) ∈ F [x] d(x) p(x) q(x)
d ′ (x) p(x) q(x) d ′ (x) | d(x) d(x) =(p(x),q(x))
p(x) q(x) (p(x),q(x)) = 1
F d(x)
p(x) q(x) F [x] r(x) s(x)
d(x) =r(x)p(x)+s(x)q(x).
d(x)
S = {f(x)p(x)+g(x)q(x) :f(x),g(x) ∈ F [x]}.
d(x) =r(x)p(x) +s(x)q(x) r(x) s(x) F [x]
d(x) p(x) q(x) d(x)

p(x) a(x) b(x) p(x) =
a(x)d(x)+b(x) b(x) b(x) < d(x)
b(x) =p(x) − a(x)d(x)
= p(x) − a(x)(r(x)p(x)+s(x)q(x))
= p(x) − a(x)r(x)p(x) − a(x)s(x)q(x)
= p(x)(1 − a(x)r(x)) + q(x)(−a(x)s(x))
p(x) q(x) S b(x)
d(x) d(x)
p(x) d(x) q(x) d(x)
p(x) q(x)
d(x) p(x) q(x) d ′ (x)
p(x) q(x) d ′ (x) | d(x) d ′ (x)
p(x) q(x) u(x) v(x)
p(x) =u(x)d ′ (x) q(x) =v(x)d ′ (x)
d(x) =r(x)p(x)+s(x)q(x)
= r(x)u(x)d ′ (x)+s(x)v(x)d ′ (x)
= d ′ (x)[r(x)u(x)+s(x)v(x)].
d ′ (x) | d(x) d(x) p(x) q(x)
p(x) q(x)
d ′ (x) p(x) q(x)
u(x) v(x) F [x] d(x) =d ′ (x)[r(x)u(x)+
s(x)v(x)]
d(x) = d ′ (x)+[r(x)u(x)+s(x)v(x)]
d(x) d ′ (x) d(x) = d ′ (x) d(x)
d ′ (x) d(x) =
d ′ (x)
f(x) ∈ F [x] F f(x)
g(x) h(x) F [x] g(x)
h(x) f(x)
x 2 − 2 ∈ Q[x]
x 2 +1
p(x) =x 3 + x 2 +2 Z 3 [x]
Z 3 [x]
x − a a Z 3 [x]
p(a) =0
p(0) = 2
p(1) = 1

p(2) = 2.
p(x) Z 3
p(x) ∈ Q[x]
p(x) = r s (a 0 + a 1 x + ···+ a n x n ),
r, s, a 0 ,...,a n a i r s
p(x) = b 0
+ b 1
x + ···+ b n
x n ,
c 0 c 1 c n
b i c i p(x)
1
p(x) = (d 0 + d 1 x + ···+ d n x n ),
c 0 ···c n
d 0 ,...,d n d d 0 ,...,d n
d
p(x) = (a 0 + a 1 x + ···+ a n x n ),
c 0 ···c n
d i = da i a i d/(c 0 ···c n )
p(x) = r s (a 0 + a 1 x + ···+ a n x n ),
(r, s) =1
p(x) ∈ Z[x] p(x)
α(x) β(x) Q[x]
α(x) β(x) p(x) p(x) =a(x)b(x) a(x) b(x)
Z[x] α(x) = a(x) β(x) = b(x)
α(x) = c 1
(a 0 + a 1 x + ···+ a m x m )= c 1
α 1 (x)
d 1 d 1
β(x) = c 2
(b 0 + b 1 x + ···+ b n x n )= c 2
β 1 (x),
d 2 d 2
a i b i
p(x) =α(x)β(x) = c 1c 2
d 1 d 2
α 1 (x)β 1 (x) = c d α 1(x)β 1 (x),
c/d c 1 /d 1 c 2 /d 2 dp(x) =
cα 1 (x)β 1 (x)
d = 1 ca m b n = 1 p(x) c = 1
c = −1 c = 1 a m = b n = 1 a m = b n = −1
p(x) =α 1 (x)β 1 (x) α 1 (x) β 1 (x) α(x) = α 1 (x)
β(x) = β 1 (x) a(x) =−α 1 (x) b(x) =−β 1 (x)
p(x) =(−α 1 (x))(−β 1 (x)) = a(x)b(x)
c = −1
d ≠1 (c, d) =1 p p | d
p ∤ c α 1 (x) a i

p ∤ a i b j β 1 (x) p ∤ b j α ′ 1 (x)
β ′ 1 (x) Z p[x] α 1 (x)
β 1 (x) p p | d α ′ 1 (x)β′ 1 (x) =0 Z p[x]
α ′ 1 (x) β′ 1 (x) Z p[x]
d =1
p(x) =x n + a n−1 x n−1 + ···+ a 0
Z a 0 ≠0 p(x) Q p(x) α Z α
a 0
p(x) a ∈ Q p(x) x − a
p(x) Z[x] α ∈ Z
a 0 /α ∈ Z α | a 0
p(x) =(x − α)(x n−1 + ···−a 0 /α).
p(x) =x 4 − 2x 3 + x +1 p(x)
Q[x] p(x) p(x) p(x) =
(x − α)q(x) q(x) p(x)
p(x) Q[x] Z
±1 p(1) = 1 p(−1) = 3
p(x)
p(x)
p(x) =(x 2 + ax + b)(x 2 + cx + d)
= x 4 +(a + c)x 3 +(ac + b + d)x 2 +(ad + bc)x + bd,
Z[x]
a + c = −2
ac + b + d =0
ad + bc =1
bd =1.
bd =1 b = d =1 b = d = −1 b = d
ad + bc = b(a + c) =1.
a+c = −2 −2b =1 b
p(x) Q
p
f(x) =a n x n + ···+ a 0 ∈ Z[x].
p | a i i =0, 1,...,n− 1 p ∤ a n p 2 ∤ a 0 f(x) Q
f(x)
Z[x]
f(x) =(b r x r + ···+ b 0 )(c s x s + ···+ c 0 )
Z[x] b r c s r,s

p ∤ a n a n = b r c s b r c s p m k
p ∤ c k
a m = b 0 c m + b 1 c m−1 + ···+ b m c 0
p p
b 0 c m m = n a i p m

〈p(x)〉 p(x) 〈p(x)〉 ⊂〈f(x)〉
〈p(x)〉
p(x) F [x] I F [x]
〈p(x)〉 I I = 〈f(x)〉 f(x) ∈ F [x]
p(x) ∈ I p(x) =f(x)g(x) g(x) ∈ F [x]
p(x) f(x) g(x) f(x)
I = F [x] g(x) f(x) I
I = 〈p(x)〉 F [x] 〈p(x)〉
ax 2 + bx + c =0
ax 3 + bx 2 + cx + d =0
ax 3 + cx + d =0.
ax 3 + bx 2 + cx + d =0.

ax 4 + bx 3 + cx 2 + dx + e =0.
3 Z 2 [x]
(5x 2 +3x − 4) + (4x 2 − x +9) Z 12
(5x 2 +3x − 4)(4x 2 − x +9) Z 12
(7x 3 +3x 2 − x)+(6x 2 − 8x +4) Z 9
(3x 2 +2x − 4) + (4x 2 +2) Z 5
(3x 2 +2x − 4)(4x 2 +2) Z 5
(5x 2 +3x − 2) 2 Z 12
q(x) r(x) a(x) =q(x)b(x)+r(x)
r(x) < b(x)
a(x) =5x 3 +6x 2 − 3x +4 b(x) =x − 2 Z 7 [x]
a(x) =6x 4 − 2x 3 + x 2 − 3x +1 b(x) =x 2 + x − 2 Z 7 [x]
a(x) =4x 5 − x 3 + x 2 +4 b(x) =x 3 − 2 Z 5 [x]
a(x) =x 5 + x 3 − x 2 − x b(x) =x 3 + x Z 2 [x]
p(x) q(x)
d(x) = (p(x),q(x)) a(x) b(x)
a(x)p(x)+b(x)q(x) =d(x)
p(x) =x 3 − 6x 2 +14x − 15 q(x) =x 3 − 8x 2 +21x − 18 p(x),q(x) ∈ Q[x]
p(x) =x 3 + x 2 − x +1 q(x) =x 3 + x − 1 p(x),q(x) ∈ Z 2 [x]
p(x) =x 3 + x 2 − 4x +4 q(x) =x 3 +3x − 2 p(x),q(x) ∈ Z 5 [x]
p(x) =x 3 − 2x +4 q(x) =4x 3 + x +3 p(x),q(x) ∈ Q[x]
5x 3 +4x 2 − x +9 Z 12
3x 3 − 4x 2 − x +4 Z 5
5x 4 +2x 2 − 3 Z 7
x 3 + x +1 Z 2

Z[x]
p(x) Z 4 [x] p(x) > 1
Q[x]
x 4 − 2x 3 +2x 2 + x +4
x 4 − 5x 3 +3x − 2
3x 5 − 4x 3 − 6x 2 +6
5x 5 − 6x 4 − 3x 2 +9x − 15
2 3 Z 2 [x]
x 2 + x +8 Z 10 [x]
p(x) Z 6 [x] n
n
F F [x 1 ,...,x n ]
Z[x]
x p + a a ∈ Z p p
f(x) F [x] F f(x) | p(x)q(x)
f(x) | p(x) f(x) | q(x)
R S R[x] ∼ = S[x]
F a ∈ F p(x) ∈ F [x] p(a)
p(x) x − a
p(x) =a n x n + a n−1 x n−1 + ···+ a 0 ∈ Z[x],
a n ≠0 p(r/s) =0 (r, s) =1 r | a 0 s | a n
Q ∗ Q ∗
(Z[x], +)
Φ n (x) = xn − 1
x − 1 = xn−1 + x n−2 + ···+ x +1
Φ p (x) Q
p
F F [x]
R
R[x]
R
R[x]
x p − x p Z p p
x p − x = x(x − 1)(x − 2) ···(x − (p − 1)).
F f(x) =a 0 + a 1 x + ··· + a n x n F [x] f ′ (x) =
a 1 +2a 2 x + ···+ na n x n−1 f(x)
(f + g) ′ (x) =f ′ (x)+g ′ (x).
D : F [x] → F [x]
D(f(x)) = f ′ (x)

D F =0
D F = p
(fg) ′ (x) =f ′ (x)g(x)+f(x)g ′ (x).
f(x) ∈ F [x]
f(x) =a(x − a 1 )(x − a 2 ) ···(x − a n ).
f(x) f(x) f ′ (x)
F F [x]
R R[x 1 ,...,x n ]
R R[x] R ′
R
p(x) q(x) R[x] R
(p(x)+q(x)) ≤ ( p(x), q(x))
ax 2 + bx + c =0
x = −b ± √ b 2 − 4ac
.
2a
Δ=b 2 − 4ac
Δ > 0 Δ=0
Δ < 0
x 3 + bx 2 + cx + d =0
y 3 + py + q =0 x = y − b/3
ω = −1+i√ 3
2
ω 2 = −1 − i√ 3
2
ω 3 =1.
y = z − p
3z

y y 3 + py + q =0 A B z 3
−p 3 /27
3√
AB = −p/3
z
3√
A, ω
3 √ A, ω 2 3 √ A,
3√
B, ω
3 √ B, ω 2 3 √ B
y
√
ω i 3 − q √
√
p
2 + 3
27 + q2
4 + ω2i 3 − q √
p
2 − 3
27 + q2
4 ,
i =0, 1, 2
y 3 + py + q =0
Δ= p3
27 + q2
4 .
Δ=0
Δ > 0
Δ < 0
x 3 − 4x 2 +11x +30=0
x 3 − 3x +5=0
x 3 − 3x +2=0
x 3 + x +3=0
x 4 + ax 3 + bx 2 + cx + d =0
y 4 + py 2 + qy + r =0
x = y − a/4
(
y 2 + 1 ) 2 ( ) 1
2 z =(z − p)y 2 − qy +
4 z2 − r .
(my + k) 2
( ) 1
q 2 − 4(z − p)
4 z2 − r =0.
z 3 − pz 2 − 4rz +(4pr − q 2 )=0.

(
y 2 + 1 ) 2
2 z =(my + k) 2
x 4 − x 2 − 3x +2=0
x 4 + x 3 − 7x 2 − x +6=0
x 4 − 2x 2 +4x − 3=0
x 4 − 4x 3 +3x 2 − 5x +2=0

18
Z[x]
Q
Z
Q
D F
D
p/q ∈ Q p q
1/2 = 2/4 = 3/6
a
b = c d
ad = bc
Q Z × Z
p/q (p, q) (3, 7)
3/7 Z × Z
5/0 (5, 0) (3, 6) (2, 4)
1/2
(a, b) (c, d)
ad = bc
D
S = {(a, b) :a, b ∈ D b ≠0}.
S (a, b) ∼ (c, d) ad = bc
∼ S

D ab = ba ∼ D
(a, b) ∼ (c, d) ad = bc cb = da (c, d) ∼ (a, b)
(a, b) ∼ (c, d) (c, d) ∼
(e, f) ad = bc cf = de ad = bc f
afd = adf = bcf = bde = bed.
D af = be (a, b) ∼ (e, f)
S F D
F D
Q
a
b + c ad + bc
= ;
d bd
a
b · c
d = ac
bd .
F D
(a, b) ∈ S [a, b]
F D
[a, b]+[c, d] =[ad + bc, bd]
[a, b] · [c, d] =[ac, bd],
F D
[a 1 ,b 1 ]=[a 2 ,b 2 ] [c 1 ,d 1 ]=[c 2 ,d 2 ]
[a 1 d 1 + b 1 c 1 ,b 1 d 1 ]=[a 2 d 2 + b 2 c 2 ,b 2 d 2 ]
(a 1 d 1 + b 1 c 1 )(b 2 d 2 )=(b 1 d 1 )(a 2 d 2 + b 2 c 2 ).
[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
(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
= a 1 b 2 d 1 d 2 + b 1 b 2 c 1 d 2
= b 1 a 2 d 1 d 2 + b 1 b 2 d 1 c 2
=(b 1 d 1 )(a 2 d 2 + b 2 c 2 ).
S F D ∼
[a, b]+[c, d] =[ad + bc, bd]
[a, b] · [c, d] =[ac, bd],

[0, 1] [1, 1]
[0, 1]
[a, b]+[0, 1] = [a1+b0,b1] = [a, b].
[1, 1] [a, b] ∈ F D a ≠0
[b, a] F D [a, b] · [b, a] =[1, 1] [b, a]
[a, b] [−a, b] [a, b]
F D
F D
F D
[a, b][e, f]+[c, d][e, f] =[ae, bf]+[ce, df]
=[aedf + bfce, bdf 2 ]
=[aed + bce, bdf]
=[ade + bce, bdf]
=([a, b]+[c, d])[e, f]
F D
D
D D
F D F D
D F D E
D ψ : F D → E
E ψ(a) =a a ∈ D a F D
D F D
φ : D → F D φ(a) =[a, 1] a b D
φ(a + b) =[a + b, 1] = [a, 1]+[b, 1] = φ(a)+φ(b)
φ(ab) =[ab, 1] = [a, 1][b, 1] = φ(a)φ(b);
φ φ φ(a) =φ(b)
[a, 1] = [b, 1] a = a1 =1b = b F D
D
φ(a)[φ(b)] −1 =[a, 1][b, 1] −1 =[a, 1] · [1,b]=[a, b].
E D ψ : F D → E ψ([a, b]) = ab −1
ψ [a 1 ,b 1 ]=[a 2 ,b 2 ] a 1 b 2 = b 1 a 2 a 1 b −1
1 =
a 2 b −1
2 ψ([a 1 ,b 1 ]) = ψ([a 2 ,b 2 ])
[a, b] [c, d] F D
ψ([a, b]+[c, d]) = ψ([ad + bc, bd])
=(ad + bc)(bd) −1
= ab −1 + cd −1
= ψ([a, b]) + ψ([c, d])

ψ([a, b] · [c, d]) = ψ([ac, bd])
=(ac)(bd) −1
= ab −1 cd −1
= ψ([a, b])ψ([c, d]).
ψ
ψ
ψ([a, b]) = ab −1 =0 a =0b =0 [a, b] =[0,b] ψ
[0,b] F D ψ
Q Q[x] Q[x]
p(x)/q(x) p(x) q(x)
q(x) Q(x)
F F
Q
F p F
Z p
F
F [x]
R a b R
a b a | b c ∈ R b = ac
R a b R
u R a = ub
D p ∈ D
p = ab a b p
p | ab p | a p | b
R Q[x, y] x 2 y 2 xy
R xy xy x 2 y 2
x 2 y 2
n>1
p 1 ···p k p i
p i
D D
a ∈ D a ≠0 a a
D

a = p 1 ···p r = q 1 ···q s p i q i r = s
π ∈ S r p i q π(j) j =1,...,r
Z[ √ 3 i] ={a + b √ 3 i}
z = a + b √ 3 i ν : Z[ √ 3 i] → N ∪{0} ν(z) =|z| 2 = a 2 +3b 2
ν(z) ≥ 0 z =0
ν(zw) =ν(z)ν(w) ν(z) =1 z
Z[ √ 3 i] 1 −1
4
4=2· 2=(1− √ 3 i)(1 + √ 3 i).
Z[ √ 3 i] 2
2=zw z,w Z[ √ 3 i] ν(z) =ν(w) =2
z Z[ √ 3 i] ν(z) =2
a 2 +3b 2 =2 2
1− √ 3 i 1+ √ 3 i 2
1 − √ 3 i 1+ √ 3 i 4
R
a ∈ R 〈a〉 = {ra : r ∈ R}
D a, b ∈ D
a | b 〈b〉 ⊂〈a〉
a b 〈b〉 = 〈a〉
a D 〈a〉 = D
a | b b = ax x ∈ D r
D br =(ax)r = a(xr) 〈b〉 ⊂〈a〉 〈b〉 ⊂〈a〉 b ∈〈a〉
b = ax x ∈ D a | b
a b u a = ub
b | a 〈a〉 ⊂〈b〉 〈b〉 ⊂〈a〉 〈a〉 = 〈b〉
〈a〉 = 〈b〉 a | b b | a a = bx b = ay x, y ∈ D
a = bx = ayx D xy =1 x y
a b
a ∈ D a 1 a
1 〈a〉 = 〈1〉 = D
D 〈p〉 D 〈p〉
p
〈p〉 a D p
〈p〉 ⊂〈a〉 〈p〉 D = 〈a〉 〈p〉 = 〈a〉 a p
a p
p 〈a〉 D 〈p〉 ⊂〈a〉 ⊂D
a | p p a a p
D = 〈a〉 〈p〉 = 〈a〉 〈p〉

D p p
p p | ab 〈ab〉 ⊂〈p〉
〈p〉 〈p〉 a ∈〈p〉 b ∈〈p〉
p | a p | b
D I 1 ,I 2 ,... I 1 ⊂ I 2 ⊂ ···
N I n = I N n ≥ N
I = ⋃ ∞
i=1 I i D I I 1 ⊂ I
0 ∈ I a, b ∈ I a ∈ I i b ∈ I j i j N
i ≤ j a b I j a − b
I j r ∈ D a ∈ I a ∈ I i i
I i ra ∈ I i I I
D
D a ∈ D I
a I N N ∈ N I N = I = 〈a〉 I n = I N
n ≥ N
D a D
a
a = a 1 b 1 a 1 b 1 〈a〉 ⊂〈a 1 〉
〈a〉 ≠ 〈a 1 〉 a a 1 b 1
a 1 = a 2 b 2 a 2 b 2
〈a 1 〉⊂〈a 2 〉
〈a〉 ⊂〈a 1 〉⊂〈a 2 〉⊂···.
N 〈a n 〉 = 〈a N 〉 n ≥ N
a N a
a = c 1 p 1 p 1 c 1
〈a〉 ⊂〈c 1 〉 c 1 c 1 = c 2 p 2
p 2 c 2
〈a〉 ⊂〈c 1 〉⊂〈c 2 〉⊂···.
a = p 1 p 2 ···p r
p 1 ,...,p r
a = p 1 p 2 ···p r = q 1 q 2 ···q s ,

p i q i
r

z w Q(i) ={p + qi : p, q ∈ Q} Z[i]
zw −1 =(a + bi) c − di
c 2 + d 2
ac + bd bc − ad
=
c 2 +
+ d2 c 2 + d 2 i
(
= m 1 + n ) (
1
c 2 + d 2 + m 2 + n )
2
c 2 + d 2 i
(
n1
=(m 1 + m 2 i)+
c 2 + d 2 + n )
2
c 2 + d 2 i
=(m 1 + m 2 i)+(s + ti)
Q(i)
m i
|n i /(a 2 + b 2 )|≤1/2
9
8 =1+1 8
15
8 =2− 1 8 .
s t zw −1 =(m 1 + m 2 i)+(s + ti)
s 2 + t 2 ≤ 1/4 + 1/4 = 1/2 w
z = zw −1 w = w(m 1 + m 2 i)+w(s + ti) =qw + r,
q = m 1 + m 2 i r = w(s + ti) z qw Z[i] r Z[i]
r =0 ν(r)

Z[x] p(x) =5x 4 −3x 3 +x−4
1 q(x) =
4x 2 − 6x +8 q(x) 2
D f(x) g(x)
D[x] f(x)g(x)
f(x) = ∑ m
i=0 a ix i g(x) = ∑ n
i=0 b ix i p
f(x)g(x) r p ∤ a r s
p ∤ b s x r+s f(x)g(x)
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 .
p a 0 ,...,a r−1 b 0 ,...,b s−1 p c r+s
a r b s p | c r+s p a r p b s
D p(x) q(x) D[x]
p(x)q(x) p(x) q(x)
p(x) =cp 1 (x) q(x) =dq 1 (x) c d p(x)
q(x) p 1 (x) q 1 (x) p(x)q(x) =
cdp 1 (x)q 1 (x) p 1 (x)q 1 (x) p(x)q(x) cd
D F p(x) ∈ D[x]
p(x) =f(x)g(x) f(x) g(x) F [x] p(x) =f 1 (x)g 1 (x) f 1 (x)
g 1 (x) D[x] f(x) = f 1 (x) g(x) = g 1 (x)
a b D af(x),bg(x) D[x]
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)
D[x] abp(x) =(a 1 f 1 (x))(b 1 g 1 (x)) f 1 (x)
g 1 (x) ab | a 1 b 1
c ∈ D p(x) =cf 1 (x)g 1 (x) f(x) = f 1 (x)
g(x) = g 1 (x)
D F p(x)
D[x] F [x] D[x]
D F p(x)
D[x] p(x) =f(x)g(x) F [x] p(x) =f 1 (x)g 1 (x) f 1 (x) g 1 (x)
D[x] f(x) = f 1 (x) g(x) = g 1 (x)
D D[x]
p(x) D[x] p(x)
D p(x)
D[x] F D
p(x) =f 1 (x)f 2 (x) ···f n (x) p(x) f i (x)
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)
g i (x) D[x] g i (x)
D[x]
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) =
ag 1 (x) ···g n (x) a ∈ D D a uc 1 ···c k u
c i D
p(x) =a 1 ···a m f 1 (x) ···f n (x) =b 1 ···b r g 1 (x) ···g s (x)
p(x) D[x]
f i g i F [x] a i b i
F F [x] n = s g i (x)
f i (x) g i (x) i =1,...,n c 1 ,...,c n d 1 ,...,d n
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)
c i d i D a 1 ···a m = ub 1 ···b r D u
D D m = s
b i a i b i i
F F [x]
Z[x]
D D[x 1 ,...,x n ]
Z[x]
17
n
N =2 2n +1 N
i √ −1

z = a + b √ 3 i Z[ √ 3 i] a 2 +3b 2 =1 z
Z[ √ 3 i] 1 −1
Z[i] Z[i]
5
1+3i
6+8i
2
D
F D
F D
F D
F 1
D D
D
F F
Q
F
F [x] F (x)
p(x)/q(x) q(x)
p(x 1 ,...,x n ) q(x 1 ,...,x n ) F [x 1 ,...,x n ]
p(x 1 ,...,x n )/q(x 1 ,...,x n )
F [x 1 ,...,x n ] F [x 1 ,...,x n ]
F (x 1 ,...,x n )

p Z p [x] Z p (x) Z p (x)
p
Z[i]
Q(i) ={p + qi : p, q ∈ Q}.
F E
F E E F
F F
Q
F p F
Z p
Z[ √ 2]={a + b √ 2:a, b ∈ Z}
Z[ √ 2]
Z[ √ 2]
Z[ √ 2]
Z[ √ 2i] ν(a+b √ 2 i) =
a 2 +2b 2
D d ∈ D a b
D d | a d | b d a b
D a b D
a b d d ′
a b d d ′ (a, b)
a b
D a b D
s t D (a, b) =as + bt
D D a ∼ b a b
D ∼ D
D ν u D
ν(u) =ν(1)
D ν a b
D ν(a) =ν(b)
Z[ √ 5 i]
R
a 1 ,...,a n R r ∈ R a 1 r 1 + ···+ a n r n
r 1 ,...,r n R R
R
D I 1 ⊃ I 2 ⊃ I 3 ⊃ ···
N I k = I N k ≥ N

D
R
R S 1 ∈ S ab ∈ S a, b ∈ S
∼ R × S (a, s) ∼ (a ′ ,s ′ ) s ∗ ∈ S
s ∗ (s ′ a − sa ′ )=0 ∼ R × S
a/s (a, s) ∈ R × S S −1 R
∼
S −1 R
a
s + b t
a
s
=
at + bs
st
b
t = ab
st ,
S −1 R S −1 R
S −1 R
R S
ψ : R → S −1 R ψ(a) =a/1
R 0/∈ S ψ
P R S = R \ P
R
P R S = R \ P S −1 R

19
N Z Q R
X X × X P X
X
(a, a) ∈ P a ∈ X
(a, b) ∈ P (b, a) ∈ P a = b
(a, b) ∈ P (b, c) ∈ P (a, c) ∈ P
a ≼ b (a, b) ∈ P
a ≤ b a b A ⊂ B
A B X ≼
a ≤ b
a b Z
X X
X X P(X) X = {a, b, c}
P(X) {a, b, c}
∅ {a} {b} {c}
{a, b} {a, c} {b, c} {a, b, c}.
X ⊂
{a, b, c}

{a, b, c}
{a, b} {a, c} {b, c}
{a} {b} {c}
∅
P({a, b, c})
G G
N a ≼ b a | b a | a
a ∈ N m | n n | m m = n
m | n n | p m | p
X = {1, 2, 3, 4, 6, 8, 12, 24} 24
X
24
Y X u X Y a ≼ u
a ∈ Y u Y u ≼ v
v Y u Y l
X Y l ≼ a a ∈ Y l Y
k ≼ l k Y l
Y
Y = {2, 3, 4, 6} X Y
12 24 12 1
Y X Y
Y Y Y

u 1 u 2 Y
u 1 ≼ u u Y u 1 ≼ u 2 u 2 ≼ u 1
u 1 = u 2
L
L
a, b ∈ L a b a ∨ b
a, b ∈ L a b a ∧ b
X X P(X)
A B P(X) A B A ∪ B A ∪ B
A B A ⊂ A ∪ B B ⊂ A ∪ B C
A B C A ∪ B A ∪ B A B
A B A ∩ B
G X G
X ⊂ G
H K G H K H ∩ K
H ∪ K G
H K H ∪ K
(A ∪ B) ′ A ′ ∩ B ′
≼ ≽ ∨ ∧
L ∨ ∧
a, b, c ∈ L
a ∨ b = b ∨ a a ∧ b = b ∧ a
a ∨ (b ∨ c) =(a ∨ b) ∨ c a ∧ (b ∧ c) =(a ∧ b) ∧ c
a ∨ a = a a ∧ a = a
a ∨ (a ∧ b) =a a ∧ (a ∨ b) =a
a ∨ b {a, b} b ∨ a
{b, a} {a, b} = {b, a}
a ∨ (b ∨ c) (a ∨ b) ∨ c {a, b, c}
d = a ∨ b c ≼ d ∨ c =(a ∨ b) ∨ c
a ≼ a ∨ b = d ≼ d ∨ c =(a ∨ b) ∨ c.
b ≼ (a ∨ b) ∨ c (a ∨ b) ∨ c
{a, b, c} (a ∨ b) ∨ c {a, b, c}
u {a, b, c} a ≼ u b ≼ u d = a ∨ b ≼ u

c ≼ u (a ∨ b) ∨ c = d ∨ c ≼ u (a ∨ b) ∨ c
{a, b, c} a ∨ (b ∨ c)
{a, b, c} a ∨ (b ∨ c) =(a ∨ b) ∨ c
a a {a} a ∨ a = a
d = a ∧ b a ≼ a ∨ d d = a ∧ b ≼ a a ∨ d ≼ a
a ∨ (a ∧ b) =a
L ∨ ∧
L ∨ ∧
L a ≼ b a ∨ b = b L ≼
a, b ∈ L a b a ∨ b
a ∧ b
L ≼ a∨a = a a ≼ a ≼
≼ a ≼ b b ≼ a a ∨ b = b b ∨ a = a
b = a ∨ b = b ∨ a = a ≼
a ≼ b b ≼ c a ∨ b = b b ∨ c = c
a ∨ c = a ∨ (b ∨ c) =(a ∨ b) ∨ c = b ∨ c = c,
a ≼ c
L a ∨ b a ∧ b
a b a =(a ∨ b) ∧ a = a ∧ (a ∨ b)
a ≼ a ∨ b b ≼ a ∨ b a ∨ b a b u
a b a ≼ u b ≼ u a ∨ b ≼ u
(a ∨ b) ∨ u = a ∨ (b ∨ u) =a ∨ u = u.
a ∧ b a b
P(X) X
P(X) X ∅ A
P(X) A ∩ X = A A ∪∅= A
I X a ≼ I a ∈ X
O X O ≼ a a ∈ X
A P(X) A
A ′ = X \ A = {x : x ∈ X x /∈ A}.
A ∪ A ′ = X A ∩ A ′ = ∅
L I O
a ∈ L a ′ a ∨ a ′ = I a ∧ a ′ = O
L ∨ ∧
a ∧ (b ∨ c) =(a ∧ b) ∨ (a ∧ c);

P(X)
A ∩ (B ∪ C) =(A ∩ B) ∪ (A ∩ C)
A, B, C ∈P(X) L
a ∧ (b ∨ c) =(a ∧ b) ∨ (a ∧ c)
a, b, c ∈ L
L
a, b, c ∈ L
a ∨ (b ∧ c) =(a ∨ b) ∧ (a ∨ c)
L
a ∨ (b ∧ c) =[a ∨ (a ∧ c)] ∨ (b ∧ c)
= a ∨ [(a ∧ c) ∨ (b ∧ c)]
= a ∨ [(c ∧ a) ∨ (c ∧ b)]
= a ∨ [c ∧ (a ∨ b)]
= a ∨ [(a ∨ b) ∧ c]
=[(a ∨ b) ∧ a] ∨ [(a ∨ b) ∧ c]
=(a ∨ b) ∧ (a ∨ c).
B I
O B X P(X)
∨ ∧
B
∨ ∧ B
a ∨ b = b ∨ a a ∧ b = b ∧ a a, b ∈ B
a ∨ (b ∨ c) =(a ∨ b) ∨ c a ∧ (b ∧ c) =(a ∧ b) ∧ c a, b, c ∈ B
a ∧ (b ∨ c) =(a ∧ b) ∨ (a ∧ c) a ∨ (b ∧ c) =(a ∨ b) ∧ (a ∨ c) a, b, c ∈ B
I O a ∨ O = a a ∧ I = a a ∈ B
a ∈ B a ′ ∈ B a ∨ a ′ = I a ∧ a ′ = O
B
a = a ∨ O
= a ∨ (a ∧ a ′ )
=(a ∨ a) ∧ (a ∨ a ′ )

=(a ∨ a) ∧ I
= a ∨ a.
I ∨ b =(b ∨ b ′ ) ∨ b =(b ′ ∨ b) ∨ b = b ′ ∨ (b ∨ b) =b ′ ∨ b = I.
a ∨ (a ∧ b) =(a ∧ I) ∨ (a ∧ b)
= a ∧ (I ∨ b)
= a ∧ I
= a.
B
B
B
a ∈ B O ∨ a = a O ≼ a O B
I B a ∨ b = b a ∧ b = a
a ∨ I = a a ∈ B
a ∨ I =(a ∧ I) ∨ I = I ∨ (I ∧ a) =I
a ≼ I a B B B
B I O
B a ∨ b a ∧ b
{a, b} B
B
a ∨ I = I a ∧ O = O a ∈ B
a ∨ b = a ∨ c a ∧ b = a ∧ c a, b, c ∈ B b = c
a ∨ b = I a ∧ b = O b = a ′
(a ′ ) ′ = a a ∈ B
I ′ = O O ′ = I
(a ∨ b) ′ = a ′ ∧ b ′ (a ∧ b) ′ = a ′ ∨ b ′
a ∨ b = a ∨ c a ∧ b = a ∧ c
b = b ∨ (b ∧ a)
= b ∨ (a ∧ b)
= b ∨ (a ∧ c)
=(b ∨ a) ∧ (b ∨ c)

=(a ∨ b) ∧ (b ∨ c)
=(a ∨ c) ∧ (b ∨ c)
=(c ∨ a) ∧ (c ∨ b)
= c ∨ (a ∧ b)
= c ∨ (a ∧ c)
= c ∨ (c ∧ a)
= c.
B C φ : B → C
φ(a ∨ b) =φ(a) ∨ φ(b)
φ(a ∧ b) =φ(a) ∧ φ(b)
a b B
X
B
a ∈ B B a ≠ O a ∧ b = a b ∈ B a
B b ∈ B a O ≼ b ≼ a
B b B
a B a ≼ b
b a = b b 1 O b
b 1 ≼ b b b 1
b 2 O b 1 b 2 ≼ b 1
b 2 a = b 2
O ≼···≼b 3 ≼ b 2 ≼ b 1 ≼ b.
B k b k
a = b k
a b B a ≠ b
a ∧ b = O
a ∧ b a b a ∧ b ≼ a
a ∧ b = a a ∧ b = O a ∧ b = a a ≼ b a = O
a b a ∧ b = O
B a, b ∈ B
a ≼ b
a ∧ b ′ = O

a ′ ∨ b = I
⇒ a ≼ b a ∨ b = b
a ∧ b ′ = a ∧ (a ∨ b) ′
= a ∧ (a ′ ∧ b ′ )
=(a ∧ a ′ ) ∧ b ′
= O ∧ b ′
= O.
⇒ a ∧ b ′ = O a ′ ∨ b =(a ∧ b ′ ) ′ = O ′ = I
⇒ a ′ ∨ b = I
a ≼ b
a = a ∧ (a ′ ∨ b)
=(a ∧ a ′ ) ∨ (a ∧ b)
= O ∨ (a ∧ b)
= a ∧ b.
B b c B b ⋠ c
a ∈ B a ≼ b a ⋠ c
b ∧ c ′ ≠ O a a ≼ b ∧ c ′
a ≼ b a ⋠ c
b ∈ B a 1 ,...,a n B a i ≼ b
b = a 1 ∨···∨a n a, a 1 ,...,a n B a ≼ b a i ≼ b
b = a ∨ a 1 ∨···∨a n a = a i i =1,...,n
b 1 = a 1 ∨···∨a n a i ≼ b i b 1 ≼ b
b ≼ b 1 b ⋠ b 1
a a ≼ b a ⋠ b 1 a a ≼ b
a = a i a i a ≼ b 1 b ≼ b 1
b = a 1 ∨···∨a n a b
a = a ∧ b = a ∧ (a 1 ∨···∨a n )=(a ∧ a 1 ) ∨···∨(a ∧ a n ).
O a a ∧ a i a i
a = a i i
B X
B P(X)
B P(X) X B
a ∈ B a a = a 1 ∨···∨a n a 1 ,...,a n ∈ X
φ : B →P(X)
φ(a) =φ(a 1 ∨···∨a n )={a 1 ,...,a n }.
φ
a = a 1 ∨···∨a n b = b 1 ∨···∨b m B a i
b i φ(a) =φ(b) {a 1 ,...,a n } = {b 1 ,...,b m } a = b φ
a b φ
φ(a ∨ b) =φ(a 1 ∨···∨a n ∨ b 1 ∨···∨b m )

φ(a ∧ b) =φ(a) ∩ φ(b)
= {a 1 ,...,a n ,b 1 ,...,b m }
= {a 1 ,...,a n }∪{b 1 ,...,b m }
= φ(a 1 ∨···∨a n ) ∪ φ(b 1 ∧···∨b m )
= φ(a) ∪ φ(b).
2 n
n
a
a
b
A B
a b a ∧ b
a b
A B
a b a ∨ b
A a b B
a ∧ b
a
A
B
b
a ∨ b

a∧b b∧a
∨ ∧
a ∧ (b ∨ c) =(a ∧ b) ∨ (a ∧ c)
a a ′ a
a I
O a ∧ a ′ = O a ∨ a ′ = I
b
a
b
a
c
a
c
a ∧ (b ∨ c) =(a ∧ b) ∨ (a ∧ c)
a a ′ a
a ∧ a ′ = O a ∨ a ′ = I
a ′
(a ∨ b) ∧ (a ∨ b ′ ) ∧ (a ∨ b)
(a ∨ b) ∧ (a ∨ b ′ ) ∧ (a ∨ b) =(a ∨ b) ∧ (a ∨ b) ∧ (a ∨ b ′ )
=(a ∨ b) ∧ (a ∨ b ′ )
= a ∨ (b ∧ b ′ )
= a ∨ O
= a,
a
a
a
a
b
b ′
b
(a ∨ b) ∧ (a ∨ b ′ ) ∧ (a ∨ b)

X = {a, b, c, d}
⊂
30
Z 12
B 36 B
a ≼ b a | b B X B
P(X)
Z a ≼ b a | b
(a ∨ b ∨ a ′ ) ∧ a
(a ∨ b) ′ ∧ (a ∨ b)
a ∨ (a ∧ b)
(c ∨ a ∨ b) ∧ c ′ ∧ (a ∨ b) ′
a b c

a
a b c
a c ′ a
a ′ b
b
c ′
X n P(X) =2 n
2 n n ∈ N
a ′ a b ′
b
a a b
a ′
b a ′ b
a b c
a ′ b ′ c
a b ′ c ′
a ≼ b
a | b
L ∨ ∧
L
a ≼ b a ∨ b = b a b
a ∧ b
G X G
H K G H K
H ∪ K
R X R X
⊂ I J X I ∩ J
I J I + J R

B
a ∨ I = I a ∧ O = O a ∈ B
a ∨ b = I a ∧ b = O b = a ′
(a ′ ) ′ = a a ∈ B
I ′ = O O ′ = I
(a ∨ b) ′ = a ′ ∧ b ′ (a ∧ b) ′ = a ′ ∨ b ′
B + · B
a + b =(a ∧ b ′ ) ∨ (a ′ ∧ b)
a · b = a ∧ b.
B a 2 = a a ∈ B
X a b X a ≼ b b ≼ a X
a | b N
N Z Q R ≤
X Y φ : X → Y a ≼ b
φ(a) ≼ φ(b) L M ψ : L → M
ψ(a ∨ b) =ψ(a) ∨ ψ(b) ψ(a ∧ b) =ψ(a) ∧ ψ(b)
B a = b (a ∧ b ′ ) ∨ (a ′ ∧ b) =O
a, b ∈ B
B a = O (a ∧ b ′ ) ∨ (a ′ ∧ b) =b
b ∈ B
L M L × M (a, b) ≼ (c, d) a ≼ c
b ≼ d L × M
n f : {O, I} n →{0,I}
x 1 ,...,x n O I
∨ ∧ ′
x y x ′ x ∨ y x∧ y
0 0 1 0 0
0 1 1 1 0
1 0 0 1 0
1 1 0 1 1

20
x y z
n
V F α · v αv
α ∈ F v ∈ V
α(βv) =(αβ)v
(α + β)v = αv + βv
α(u + v) =αu + αv
1v = v
α, β ∈ F u, v ∈ V
V F
n R n R
u =(u 1 ,...,u n ) v =(v 1 ,...,v n ) R n α R
u + v =(u 1 ,...,u n )+(v 1 ,...,v n )=(u 1 + v 1 ,...,u n + v n )
αu = α(u 1 ,...,u n )=(αu 1 ,...,αu n ).

F F [x] F F [x]
α ∈ F
p(x) ∈ F [x] αp(x)
[a, b]
R f(x) g(x) [a, b] (f + g)(x)
f(x) +g(x) (αf)(x) =αf(x) α ∈ R
f(x) = x g(x) =x 2 (2f +5g)(x) =2 x +5x 2
V = Q( √ 2) = {a + b √ 2:a, b ∈ Q} V
Q u = a + b √ 2 v = c + d √ 2 u + v =(a + c)+(b + d) √ 2 V
α ∈ Q αv V
V
V F
0v = v ∈ V
α = α ∈ F
αv = α =0 v =
(−1)v = −v v ∈ V
−(αv) =(−α)v = α(−v) α ∈ F v ∈ V
0v =(0+0)v =0v +0v;
+0v =0v +0v V =0v
α =0
α ≠0 αv = 1/α v =
v +(−1)v =1v +(−1)v =(1− 1)v =0v = ,
−v =(−1)v
V F W V W
V u, v ∈ W
α ∈ F u + v αv W
W R 3 W = {(x 1 , 2x 1 + x 2 ,x 1 − x 2 ):
x 1 ,x 2 ∈ R} W R 3
α(x 1 , 2x 1 + x 2 ,x 1 − x 2 )=(αx 1 ,α(2x 1 + x 2 ),α(x 1 − x 2 ))
=(αx 1 , 2(αx 1 )+αx 2 ,αx 1 − αx 2 ),
W W
u =(x 1 , 2x 1 + x 2 ,x 1 − x 2 ) v =(y 1 , 2y 1 + y 2 ,y 1 − y 2 ) W
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]
p(x) q(x) p(x)+q(x) αp(x) ∈ W
α ∈ F p(x) ∈ W
V F v 1 ,v 2 ,...,v n
V α 1 ,α 2 ,...,α n F w V
w =
n∑
α i v i = α 1 v 1 + α 2 v 2 + ···+ α n v n
i=1
v 1 ,v 2 ,...,v n
v 1 ,v 2 ,...,v n
v 1 ,v 2 ,...,v n W v 1 ,v 2 ,...,v n W
v 1 ,v 2 ,...,v n
S = {v 1 ,v 2 ,...,v n } V
S V
u v S
v i
u = α 1 v 1 + α 2 v 2 + ···+ α n v n
v = β 1 v 1 + β 2 v 2 + ···+ β n v n .
u + v =(α 1 + β 1 )v 1 +(α 2 + β 2 )v 2 + ···+(α n + β n )v n
v i α ∈ F
S
αu =(αα 1 )v 1 +(αα 2 )v 2 + ···+(αα n )v n
S = {v 1 ,v 2 ,...,v n } V
α 1 ,α 2 ...α n ∈ F α i
α 1 v 1 + α 2 v 2 + ···+ α n v n = ,
S S
S
{α 1 ,α 2 ...α n }
α 1 v 1 + α 2 v 2 + ···+ α n v n =
α 1 = α 2 = ···= α n =0
{v 1 ,v 2 ,...,v 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 .
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 ,
(α 1 − β 1 )v 1 +(α 2 − β 2 )v 2 + ···+(α n − β n )v n = .
v 1 ,...,v n α i − β i =0 i =1,...,n
{v 1 ,v 2 ,...,v n } V
v i
{v 1 ,v 2 ,...,v n }
α 1 ,...,α n
α 1 v 1 + α 2 v 2 + ···+ α n v n = ,
α i α k ≠0
v k = − α 1
v 1 −···− α k−1
α k
α k
v k−1 − α k+1
α k
v k+1 −···− α n
α k
v n .
v k = β 1 v 1 + ···+ β k−1 v k−1 + β k+1 v k+1 + ···+ β n v n .
β 1 v 1 + ···+ β k−1 v k−1 − v k + β k+1 v k+1 + ···+ β n v n = .
V n m>n
m V
{e 1 ,e 2 ,...,e n } V V {e 1 ,e 2 ,...,e n }
V
e 1 =(1, 0, 0) e 2 =(0, 1, 0) e 3 =(0, 0, 1)
R 3 R 3 (x 1 ,x 2 ,x 3 ) R 3
x 1 e 1 + x 2 e 2 + x 3 e 3 e 1 ,e 2 ,e 3
e 1 ,e 2 ,e 3
R 3 {(3, 2, 1), (3, 2, 0), (1, 1, 1)} R 3
Q( √ 2)={a+b √ 2:a, b ∈ Q} {1, √ 2 } {1+ √ 2, 1− √ 2 }
Q( √ 2)
R 3
Q( √ 2)
{e 1 ,e 2 ,...,e m } {f 1 ,f 2 ,...,f n }
V m = n

{e 1 ,e 2 ,...,e m }
n ≤ m {f 1 ,f 2 ,...,f n }
m ≤ n m = n
{e 1 ,e 2 ,...,e n } V
V n V = n
V n
S = {v 1 ,...,v n } V S
V
S = {v 1 ,...,v n } V S V
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}
{(x 1 ,x 2 ,x 3 ):x 1 − 2x 2 +2x 3 =2}
{(x 1 ,x 2 ,x 3 ):3x 1 − 2x 2 2 =0}
(x, y, z) ∈ R 3
Ax + By + Cz =0
Dx + Ey + Cz =0
R 3
W [0, 1] f(0) = 0
W C[0, 1]
V F −(αv) =(−α)v = α(−v) α ∈ F
v ∈ V
V n
S = {v 1 ,...,v n } V S
V
S = {v 1 ,...,v n } V S V
S = {v 1 ,...,v k } V kn
V W F
m n T : V → W
T (u + v) =T (u)+T (v)
T (αv) =αT (v)
α ∈ F u, v ∈ V T V W
T (T )={v ∈ V : T (v) =} V
T T
T R(V )={w ∈ W : T (v) =w v ∈
V } W
T : V → W (T )={}
{v 1 ,...,v k } T
{v 1 ,...,v k ,v k+1 ,...,v m } V {T (v k+1 ),...,T(v m )}
T T m − k

V = W T : V → W
V W n F
T : V → W {v 1 ,...,v n } V
{T (v 1 ),...,T(v n )} W
F n F n
U V W U
V U + V u + v u ∈ U
v ∈ V
U + V U ∩ V W
U + V = W U ∩ V = W
W = U ⊕ V w ∈ W
w = u + v u ∈ U v ∈ V
U k W n
V n − k W = U ⊕ V
V
U V W
(U + V )= U + V − (U ∩ V ).
V W F
V W (V,W)
F
(S + T )(v) =S(v)+T (v)
(αS)(v) =αS(v),
S, T ∈ (V,W) α ∈ F v ∈ V
V F V V ∗ = (V,F)
V v 1 ,...,v n
V v = α 1 v 1 + ···+ α n v n V
φ i : V → F φ i (v) =α i φ i V ∗
v 1 ,...,v n
{(3, 1), (2, −2)} R 2 (R 2 ) ∗
V n F V ∗∗
V ∗ v ∈ V λ v V ∗∗
v ↦→ λ v V V ∗∗

21
F
Q R
F p(x) ∈ F [x]
E F p(x)
E[x]
p(x) =x 4 − 5x 2 +6
Q[x] p(x) (x 2 − 2)(x 2 − 3)
Q[x] p(x)
p(x) =(x − √ 2)(x + √ 2)(x − √ 3)(x + √ 3).
p(x)
Q( √ 2) = {a + b √ 2:a, b ∈ Q}.
F
E F F E F
F ⊂ E
F = Q( √ 2)={a + b √ 2:a, b ∈ Q}
E = Q( √ 2+ √ 3) Q √ 2+ √ 3
E F E
F √ 2 E √ 2+ √ 3 E
1/( √ 2+ √ 3) = √ 3 − √ 2 E √ 2+ √ 3
√
3 −
√
2
√
2
√
3 E
p(x) =x 2 + x +1 ∈ Z 2 [x]
p(x) Z 2
Z 2 α p(α) =0 〈p(x)〉

p(x) Z 2 [x]/〈p(x)〉
Z 2 [x]/〈p(x)〉
f(x) +〈p(x)〉
f(x) =(x 2 + x +1)q(x)+r(x),
r(x) x 2 + x +1
f(x)+〈x 2 + x +1〉 = r(x)+〈x 2 + x +1〉.
r(x) 0 1 x 1+x E = Z 2 [x]/〈x 2 +x+1〉
Z 2 α p(x)
Z 2 (α)
0+0α =0
1+0α =1
0+1α = α
1+1α =1+α.
α 2 + α +1=0 (1 + α) 2
(1 + α)(1 + α) =1+α + α +(α) 2 = α.
E
+ 0 1 α 1+α
0 0 1 α 1+α
1 1 0 1+α α
α α 1+α 0 1
1+α 1+α α 1 0
Z 2 (α)
· 0 1 α 1+α
0 0 0 0 0
1 0 1 α 1+α
α 0 α 1+α 1
1+α 0 1+α 1 α
Z 2 (α)
F p(x) F [x]
E F α ∈ E p(α) =0

p(x)
E F α p(α) =0 〈p(x)〉
p(x) F [x] F [x]/〈p(x)〉
E = F [x]/〈p(x)〉
E F
ψ : F → F [x]/〈p(x)〉 ψ(a) =a + 〈p(x)〉 a ∈ F
ψ
ψ(a)+ψ(b) =(a + 〈p(x)〉)+(b + 〈p(x)〉) =(a + b)+〈p(x)〉 = ψ(a + b)
ψ(a)ψ(b) =(a + 〈p(x)〉)(b + 〈p(x)〉) =ab + 〈p(x)〉 = ψ(ab).
ψ
a + 〈p(x)〉 = ψ(a) =ψ(b) =b + 〈p(x)〉.
a−b p(x) 〈p(x)〉 p(x)
a − b =0 a = b ψ
ψ F {a + 〈p(x)〉 : a ∈ F } E
E F
p(x) α ∈ E α = x + 〈p(x)〉 α
E p(x) =a 0 + a 1 x + ···+ a n x n
p(α) =a 0 + a 1 (x + 〈p(x)〉)+···+ a n (x + 〈p(x)〉) n
= a 0 +(a 1 x + 〈p(x)〉)+···+(a n x n + 〈p(x)〉)
= a 0 + a 1 x + ···+ a n x n + 〈p(x)〉
=0+〈p(x)〉.
α ∈ E = F [x]/〈p(x)〉 α p(x)
p(x) =x 5 +x 4 +1 ∈ Z 2 [x] p(x) x 2 +x+1
x 3 + x +1 E Z 2 p(x) E E
Z 2 [x]/〈x 2 + x +1〉 Z 2 [x]/〈x 3 + x +1〉
Z 2 [x]/〈x 3 + x +1〉 2 3 =8
α E F F f(α) =0
f(x) ∈ F [x] E F
F E F F
E F E F α 1 ,...,α n E
F α 1 ,...,α n F (α 1 ,...,α n ) E = F (α)
α ∈ E E F
√ 2 i Q
x 2 −2 x 2 +1 π e
Q R
Q Q
π + e
[0, 1]

Q
C Q
√ 2+ √ 3 Q α = √ 2+ √ 3
α 2 =2+ √ 3 α 2 − 2= √ 3 (α 2 − 2) 2 =3 α 4 − 4α 2 +1=0
α x 4 − 4x 2 +1∈ Q[x]
E F E
F
E F α ∈ E α
F F (α) F (x) F [x]
φ α : F [x] → E α α
F φ α (p(x)) = p(α) ≠ 0
p(x) ∈ F [x] φ α = {0} φ α
E F [x] F [x]
F (x) E
E F α ∈ E α
F p(x) ∈ F [x]
p(α) =0f(x) F [x] f(α) =0 p(x)
f(x)
φ α : F [x] → E φ α
p(x) ∈ F [x] p(x) ≥ 1
F [x] α 〈p(x)〉
F [x] α f(α) =0 f(x)
f(x) ∈〈p(x)〉 p(x) f(x) p(x)
α α
βp(x) β ∈ F
p(x) =r(x)s(x) p(x)
p(α) =0 r(α)s(α) =0 r(α) =0 s(α) =0
p p(x)
E F α ∈ E F
p(x) α F
p(x) α F
f(x) =x 2 − 2 g(x) =x 4 − 4x 2 +1
√ 2 √ 2+ √ 3
E F α ∈ E F
F (α) ∼ = F [x]/〈p(x)〉 p(x) α F
φ α : F [x] → E
〈p(x)〉 p(x) α
φ α E F (α) F α
E = F (α) F α ∈ E
F α F n β ∈ E
β = b 0 + b 1 α + ···+ b n−1 α n−1
b i ∈ F

φ α (F [x]) ∼ = F (α) E = F (α)
φ α (f(x)) = f(α) f(α) α F
p(x) =x n + a n−1 x n−1 + ···+ a 0
α p(α) =0
α n+1 = αα n
α n = −a n−1 α n−1 −···−a 0 .
= −a n−1 α n − a n−2 α n−1 −···−a 0 α
= −a n−1 (−a n−1 α n−1 −···−a 0 ) − a n−2 α n−1 −···−a 0 α.
α m m ≥ n
α n β ∈ F (α)
b i c i F
β = b 0 + b 1 α + ···+ b n−1 α n−1 .
β = b 0 + b 1 α + ···+ b n−1 α n−1 = c 0 + c 1 α + ···+ c n−1 α n−1
g(x) =(b 0 − c 0 )+(b 1 − c 1 )x + ···+(b n−1 − c n−1 )x n−1
F [x] g(α) =0 g(x) p(x)
α g(x)
b 0 − c 0 = b 1 − c 1 = ···= b n−1 − c n−1 =0,
b i = c i i =0, 1,...,n− 1
x 2 +1 R 〈x 2 +1〉 R[x]
E = R[x]/〈x 2 +1〉 R x 2 +1 α = x+〈x 2 +1〉
E E
R(α) ={a + bα : a, b ∈ R} α 2 = −1 E
α 2 +1=(x + 〈x 2 +1〉) 2 +(1+〈x 2 +1〉)
=(x 2 +1)+〈x 2 +1〉
=0.
R(α) C a + bα
a + bi
E F E F
E F
E
E F
E F F

E = F (α) F
{1,α,α 2 ,...,α n−1 }
E F F
n E n F
E F
[E : F ]=n.
E F
α ∈ E [E : F ]=n
1,α,...,α n
a i ∈ F
p(α) =0
a n α n + a n−1 α n−1 + ···+ a 1 α + a 0 =0.
p(x) =a n x n + ···+ a 0 ∈ F [x]
F
R Q Q
E F K E K
F
[K : F ]=[K : E][E : F ].
{α 1 ,...,α n } E F {β 1 ,...,β m }
K E {α i β j } K F
K u ∈ K u = ∑ m
j=1 b jβ j b j = ∑ n
i=1 a ijα i
b j ∈ E a ij ∈ F
(
m∑ n∑
)
u = a ij α i β j = ∑ a ij (α i β j ).
j=1 i=1
i,j
mn α i β j K F
{α i β j }
v 1 ,v 2 ,...,v n V
c 1 v 1 + c 2 v 2 + ···+ c n v n =0
c 1 = c 2 = ···= c n =0.
u = ∑ c ij (α i β j )=0
i,j

c ij ∈ F c ij u
m∑
j=1
( n∑
i=1
c ij α i
)
β j =0,
∑ i c ijα i ∈ E β j E
n∑
c ij α i =0
i=1
j α j F c ij =0 i
j
F i i =1,...,k F i+1 F i
F k F 1
[F k : F 1 ]=[F k : F k−1 ] ···[F 2 : F 1 ].
E F α ∈ E F
p(x) β ∈ F (α) q(x) q(x)
p(x)
p(x) =[F (α) :F ] q(x) =[F (β) :F ] F ⊂
F (β) ⊂ F (α)
[F (α) :F ]=[F (α) :F (β)][F (β) :F ].
Q √ 3+ √ 5
√ 3+ √ 5 x 4 − 16x 2 +4
[Q( √ 3+ √ 5):Q] =4.
{1, √ 3 } Q( √ √ √
3) Q 3+ 5
Q( √ 3) √ 5 Q( √ 3) {1, √ 5 }
Q( √ 3, √ 5) = (Q( √ 3 ))( √ 5) Q( √ 3) {1, √ 3, √ 5, √ 3 √ 5= √ 15 }
Q( √ 3, √ 5) = Q( √ 3+ √ 5) Q
F (α 1 ,...,α n ) F n>1
Q( 3√ 5, √ 5 i) √ 5
5 3√ 5 5 √ 5 i /∈ Q( 3√ 5)
[Q( 3√ 5, √ 5 i) :Q( 3√ 5)]=2.
{1, √ 5i } Q( 3√ 5, √ 5 i) Q( 3√ 5)
{1, 3√ 5, ( 3√ 5) 2 } Q( 3√ 5) Q Q( 3√ 5, √ 5 i) Q
{1, √ 5 i, 3√ 5, ( 3√ 5) 2 , ( 6√ 5) 5 i, ( 6√ 5) 7 i =5 6√ 5 i 6√ 5 i}.
6√ 5 i x 6 +5
Q p =5
Q ⊂ Q( 6√ 5 i) ⊂ Q( 3√ 5, √ 5 i).
Q( 6√ 5 i) =Q( 3√ 5, √ 5 i)
6

E F
E F
α 1 ,...,α n ∈ E E =
F (α 1 ,...,α n )
E = F (α 1 ,...,α n ) ⊃ F (α 1 ,...,α n−1 ) ⊃···⊃F (α 1 ) ⊃ F,
F (α 1 ,...,α i ) F (α 1 ,...,α i−1 )
⇒ E F E
F α 1 ,...,α n
E E = F (α 1 ,...,α n ) α i F
⇒ E = F (α 1 ,...,α n ) α i F
E = F (α 1 ,...,α n ) ⊃ F (α 1 ,...,α n−1 ) ⊃···⊃F (α 1 ) ⊃ F,
F (α 1 ,...,α i ) F (α 1 ,...,α i−1 )
⇒
E = F (α 1 ,...,α n ) ⊃ F (α 1 ,...,α n−1 ) ⊃···⊃F (α 1 ) ⊃ F,
F (α 1 ,...,α i ) F (α 1 ,...,α i−1 )
F (α 1 ,...,α i )=F (α 1 ,...,α i−1 )(α i )
α i F (α 1 ,...,α i−1 )
[F (α 1 ,...,α i ):F (α 1 ,...,α i−1 )]
i [E : F ]
F E
p(x) E
E F
F
E
α, β ∈ E F F (α, β) F
F (α, β) F α ± β αβ α/β β ≠0
F E F
Q
E F F
E E F F
F [x] F
F
F [x] F [x]

F p(x) ∈ F [x]
p(x) F α x − α p(x)
p(x) =(x − α)q 1 (x) q 1 (x) = p(x) − 1 q 1 (x)
p(x) =(x − α)(x − β)q 2 (x),
q 2 (x) = p(x) − 2 p(x)
p(x) F [x]
ax − b p(b/a) =0 F
F E
E F F ⊂ E α ∈ E
α x − α α ∈ F F = E
F
F p(x) F [x]
F p(x)
E F p(x)
F p(x)
p(x)
F p(x) =a 0 + a 1 x + ···+ a n x n F [x]
E F p(x) α 1 ,...,α n
E E = F (α 1 ,...,α n )
p(x) =(x − α 1 )(x − α 2 ) ···(x − α n ).
p(x) ∈ F [x] E E[x]
p(x) =x 4 +2x 2 − 8 Q[x] p(x)
x 2 − 2 x 2 +4 Q( √ 2,i) p(x)
p(x) =x 3 − 3 Q[x] p(x) Q( 3√ 3)
p(x)
Q( 3√ 3)
− 3√ 3 ± ( 6√ 3) 5 i
,
2

p(x) ∈ F [x]
E p(x)
p(x) p(x) =1
p(x) E = F
k 1 ≤ k

E[x]/〈p(x)〉
σ
E(α)
ψ
φ
F [x]/〈q(x)〉
τ
F (β)
E
φ
F
φ : E → F p(x)
E[x] q(x) F [x]
K p(x) L q(x) φ
ψ : K → L
p(x)
p(x) E q(x) F p(x) =1
K = E L = F
n K
p(x) p(x) K α
E ⊂ E(α) ⊂ K β q(x) L F ⊂ F (β) ⊂ L
φ : E(α) → F (β) φ(α) =β φ
φ E
K
ψ
L
E(α)
φ
F (β)
E
φ
F
p(x) =(x − α)f(x) q(x) =(x − β)g(x) f(x)
g(x) p(x) q(x) K

f(x) E(α) L g(x) F (β)
ψ : K → L ψ φ
E(α) ψ : K → L ψ φ E
p(x) F [x] K
p(x)
30 ◦ 90 ◦
20 ◦
60 ◦
α |α|
F
α β α + β α − β αβ
α/β β ≠0 α β
α>β α + β α − β
αβ β>1
△ABC △ADE α/1 = x/β x
αβ β

D
β
B
1
α
A
x
C
E
α √ α
△ABD △BCD △ABC
1/x = x/α x 2 = α
B
x
A
D
α
C
P =(p, q)
p q
F R
F ax + by + c =0 a
b c F
F
F x 2 + y 2 + dx + ey + f =0 d e f F
(x 1 ,y 1 ) (x 2 ,y 2 ) F
x 1 = x 2 x − x 1 =0
ax + by + c =0 x 1 ≠ x 2
( )
y2 − y 1
y − y 1 =
(x − x 1 ),
x 2 − x 1
(x 1 ,y 1 )
r
(x − x 1 ) 2 +(y − y 1 ) 2 − r 2 =0.

F
R
R
F
F
F F
R
R
F F
R
ax + by + c =0 F F
x 2 + y 2 + d 1 x + e 1 y + f 1 =0
x 2 + y 2 + d 2 x + e 2 y + f 2 =0
d i e i f i F i =1, 2
x 2 + y 2 + d 1 x + e 1 x + f 1 =0
(d 1 − d 2 )x + b(e 2 − e 1 )y +(f 2 − f 1 )=0.
ax + by + c =0
x 2 + y 2 + dx + ey + f =0.
y Ax 2 +Bx+C =0
A B C F x
x = −B ± √ B 2 − 4AC
2A
F ( √ α ) α = B 2 − 4AC > 0
F
F F ( √ α ) α F
α
Q = F 0 ⊂ F 1 ⊂···⊂F k
F i = F i−1 ( √ α i ) α i ∈ F i α ∈ F k
k>0 [Q(α) :Q] =2 k

F i α i
[F k : Q] =[F k : F k−1 ][F k−1 : F k−2 ] ···[F 1 : Q] =2 k .
Q
1
1 2
3√ 2
3√
2 x 3 − 2 Q
[Q( 3√ 2):Q] =3
3 2
1
π √ π
π √ π
20 ◦ 60 ◦
3θ = (2θ + θ)
= 2θ θ − 2θ θ
=(2 2 θ − 1) θ − 2 2 θ θ
=(2 2 θ − 1) θ − 2(1 − 2 θ) θ
=4 3 θ − 3 θ.
θ α = θ θ =20 ◦
3θ = 60 ◦ = 1/2
4α 3 − 3α = 1 2 .
α 8x 3 − 6x − 1 Z[x]
Q[x] [Q(α) :Q] =3 α

x 2 + y 2 = z 2
x n + y n = z n n ≥ 3
p(x, y) Z[x, y]
Q
Q
√
1/3 + √ 7
√ 3+ 3√ 5
√ 3+ √ 2 i
θ + i θ θ =2π/n n ∈ N
√ 3 √ 2 − i

Q( √ 3, √ 6) Q
Q( 3√ 2, 3√ 3) Q
Q( √ 2,i) Q
Q( √ 3, √ 5, √ 7) Q
Q( √ 2, 3√ 2) Q
Q( √ 8) Q( √ 2)
Q(i, √ 2+i, √ 3+i) Q
Q( √ 2+ √ 5) Q( √ 5)
Q( √ 2, √ 6+ √ 10 ) Q( √ 3+ √ 5)
x 4 − 10x 2 +21 Q
x 4 +1 Q
x 3 +2x +2 Z 3
x 3 − 3 Q
Q( 4√ 3,i) Q
Q( 4√ 3,i) Q [Q( 4√ 3,i):Q] =8
F Q( 4√ 3,i) [F : Q] =2
F Q( 4√ 3,i) [F : Q] =4
Z 2 [x]/〈x 3 + x +1〉
9
20
1 ◦ Q
Q( √ 3, 4√ 3, 8√ 3,...) Q
π Q(π 3 )
p(x) n F [x]
E p(x) [E : F ] ≤ n!
Q( √ 2) ∼ = Q( √ 3)
Q( 4√ 3) Q( 4√ 3 i)
K E E F
K F
Z[x]/〈x 3 − 2〉
F p p(x) =x p − a
F F

E F p(x) F [x]
E
p(x) F [x] F
α β β ≠0 α/β
R Q
Q
E F σ E
F α ∈ E σ
α E
Q( √ 3, √ 7)=Q( √ 3+ √ 7) Q( √ a, √ b )=
Q( √ a + √ b ) (a, b) =1
E F [E : F ]=2 E
F f(x) ∈ F [x]
p(x) Z 6 [x]
R p(x) R
E F α ∈ E [F (α) :F (α 3 )]
α, β Q αβ α+β
E F α ∈ E F
F (α) F F
α p(x) ∈ F [x] p = n
[F (α) :F ]=n

22
Z p p
p n p n
F p p
α F pα =0 F
0 p F
n nα =0 α F
F p p n
F F p p
p
F p F p n
n ∈ N
φ : Z → F φ(n) =n · 1
F p φ pZ φ
F Z p K F
K K
[F : K] =n F F K
α 1 ,...,α n ∈ F α F
α = a 1 α 1 + ···+ a n α n ,
a i K p K p n
α i F p n
p D
p
a pn + b pn =(a + b) pn
n

n
n =1
p∑
( p
(a + b) p = a
k)
k b p−k .
0

p n
F p n p n
x pn − x Z p
f(x) =x pn − x F f(x)
f(x) p n F f ′ (x) =p n x pn−1 − 1=−1
f(x) f(x) F
f(x) α β f(x) α + β αβ f(x)
α pn +β pn =(α+β) pn α pn β pn =(αβ) pn
f(x) f(x) α f(x)
−α f(x)
f(−α) =(−α) pn − (−α) =−α pn + α = −(α pn − α) =0,
p p =2
f(−α) =(−α) 2n − (−α) =α + α =0.
α ≠0 (α −1 ) pn =(α pn ) −1 = α −1 f(x) F
f(x) F
E p n E F
E f(x) f(x) α
E E p n − 1
α pn−1 =1 α pn − α =0 E p n E
f(x)
p n p n
(p n )
(p n ) p m m
n m | n m>0 (p n )
(p m )
F E = (p n ) F K
p m K Z p m | n [E : K] =[E : F ][F :
K]
m | n m>0 p m − 1 p n − 1
x pm−1 − 1 x pn−1 − 1 x pm − x x pn − x
x pm − x x pn − x (p n )
x pm − x (p m )
(p 24 )

(p 24 )
(p 8 )
(p 12 )
(p 4 )
(p 6 )
(p 2 )
(p 3 )
(p)
(p 24 )
F F
F ∗
G F ∗
F G
G F ∗ n
G ∼ = Z e p 1 ×···×Z e
1 p k ,
k
n = p e 1
1 ···pe k
k
p 1 ,...,p k m
p e 1
1 ,...,pe k
k
G m
α G x r − 1 r m α x m − 1 x m − 1
m F n ≤ m m ≤|G|
m = n G n
E F F
α E ∗ E
E = F (α)
(2 4 ) Z 2 /〈1+x + x 4 〉
(2 4 )
{a 0 + a 1 α + a 2 α 2 + a 3 α 3 : a i ∈ Z 2 1+α + α 4 =0}.
1+α + α 4 =0 (2 4 )
(2 4 ) Z 15
α
α 1 = α α 6 = α 2 + α 3 α 11 = α + α 2 + α 3
α 2 = α 2 α 7 =1+α + α 3 α 12 =1+α + α 2 + α 3
α 3 = α 3 α 8 =1+α 2 α 13 =1+α 2 + α 3
α 4 =1+α α 9 = α + α 3 α 14 =1+α 3
α 5 = α + α 2 α 10 =1+α + α 2 α 15 =1.

(n, k)
E : Z k 2 → Zn 2 D : Zn 2 →
Z k 2 D
H ∈ M k×n (Z 2 )
φ : Z k 2 → Zn 2
(n, k) φ (a 1 ,a 2 ,...,a n )
n (a n ,a 1 ,a 2 ,...,a n−1 )
(6, 3)
⎛ ⎞
1 0 0
0 1 0
0 0 1
G 1 =
1 0 0
⎜ ⎟
⎝0 1 0⎠
0 0 1
⎛ ⎞
1 0 0
1 1 0
1 1 1
G 2 =
.
1 1 1
⎜ ⎟
⎝0 1 1⎠
0 0 1
(000) ↦→ (000000) (100) ↦→ (100100)
(001) ↦→ (001001) (101) ↦→ (101101)
(010) ↦→ (010010) (110) ↦→ (110110)
(011) ↦→ (011011) (111) ↦→ (111111).
(000) ↦→ (000000) (100) ↦→ (111100)
(001) ↦→ (001111) (101) ↦→ (110011)
(010) ↦→ (011110) (110) ↦→ (100010)
(011) ↦→ (010001) (111) ↦→ (101101).
(101101) (011011)
Z 2 n
Z 2 [x] n (a 0 ,a 1 ,...,a n−1 )
f(x) =a 0 + a 1 x + ···+ a n−1 x n−1 ,
f(x) n − 1
5 (10011)
1+0x +0x 2 +1x 3 +1x 4 =1+x 3 + x 4 .

f(x) ∈ Z 2 [x] f(x)

f(t) t
R n
C Z n 2 R n =
Z[x]/〈x n − 1〉
C f(t) C tf(t)
C t k f(t) C k ∈ N C
f(t),tf(t),t 2 f(t),...,t n−1 f(t)
p(t) p(t)f(t) C C
C Z 2 [x]/〈x n +1〉 f(t) =a 0 + a 1 t + ···+
a n−1 t n−1 C tf(t) C (a 1 ,...,a n−1 ,a 0 )
C
R n
Z n 2 R n
φ : Z 2 [x] → R n φ[f(x)] = f(t)
φ x n − 1 C R n
φ(I) I Z 2 [x] 〈x n − 1〉
I Z 2 [x] Z 2 I = 〈g(x)〉
Z 2 [x] 〈x n − 1〉 I
g(x) x n − 1 C R n
C = 〈g(t)〉 = {f(t)g(t) :f(t) ∈ R n g(x) | (x n − 1) Z 2 [x]}.
C
C
x 7 − 1
x 7 − 1=(1+x)(1 + x + x 3 )(1 + x 2 + x 3 ).
g(t) =(1+t + t 3 ) C R 7 (7, 4)
g(t)
t t 2 t 3 C
⎛ ⎞
1 0 0 0
1 1 0 0
0 1 1 0
G =
1 0 1 1
.
⎜
0 1 0 1
⎟
⎝0 0 1 0⎠
0 0 0 1
(n, k) C
t k x n − 1=g(x)h(x) Z 2 [x] g(x) =g 0 + g 1 x +
···+ g n−k x n−k h(x) =h 0 + h 1 x + ···+ h k x k n × k
⎛
⎞
g 0 0 ··· 0
g 1 g 0 ··· 0
G =
g n−k g n−k−1 ··· g 0
0 g n−k ··· g 1
⎜
⎝ ⎟ ⎠
0 0 ··· g n−k

C g(t)
C (n − k) × n
⎛
⎞
0 ··· 0 0 h k ··· h 0
0 ··· 0 h
H =
k ··· h 0 0
⎜
⎟
⎝··· ··· ··· ··· ··· ··· ··· ⎠ .
h k ··· h 0 0 0 ··· 0
C = 〈g(t)〉 R n x n − 1=
g(x)h(x) G H C
HG =0
x 7 − 1=g(x)h(x) =(1+x + x 3 )(1 + x + x 2 + x 4 ).
⎛
0 0 1 0 1 1
⎞
1
H = ⎝0 1 0 1 1 1 0⎠ .
1 0 1 1 1 0 0
α 1 ,...,α n F
n × n
⎛
⎞
1 1 ··· 1
α 1 α 2 ··· α n
α 1 2 α2 2 ··· αn
2 ⎜
⎝ ⎟ ⎠
α1 n−1 α2 n−1 ··· αn
n−1
α 1 ,...,α n F n ≥ 2
⎛
⎞
1 1 ··· 1
α 1 α 2 ··· α n
α 1 2 α2 2 ··· αn
2 = ∏
(α i − α j ).
⎜
⎝ ⎟
1≤j

p(x)
n − 1 p(x) α 1 ,...,α n−1
p(x) =(x − α 1 )(x − α 2 ) ···(x − α n−1 )β,
⎛
⎞
1 1 ··· 1
α 1 α 2 ··· α n−1
β =(−1) n+n
α 1 2 α2 2 ··· αn−1
2 .
⎜
⎝ ⎟ ⎠
α1 n−2 α2 n−2 ··· αn−1
n−2
β =(−1) n+n
∏
1≤j

(ω i 0
) r+1 x 0 +(ω i 1
) r+1 x 1 + ···+(ω i s−1
) r+1 x n−1 =0
(ω i 0
) r+s−1 x 0 +(ω i 1
) r+s−1 x 1 + ···+(ω i s−1
) r+s−1 x n−1 =0.
⎛
(ω i 0
) r (ω i 1
) r ··· (ω i s−1
) r ⎞
(ω i 0
) r+1 (ω i 1
) r+1 ··· (ω i s−1
) r+1
⎜
⎝
⎟
⎠
(ω i 0
) r+s−1 (ω i 1
) r+s−1 ··· (ω i s−1
) r+s−1
a i0 = a i1 = ···= a is−1 =0
231 24
231 + 24 = 255 = 2 8 − 1 (255, 231)
1 16
d =2r +1
r ≥ 0 ω n Z 2 m i (x)
Z 2 ω i
g(x) =[m 1 (x),m 2 (x),...,m 2r (x)],
〈g(t)〉 R n n d
C d
C = 〈g(t)〉 R n
C d
f(t) C f(ω i )=0 1 ≤ i

m i (x) m i (x) ω i g(x) | f(x)
g(x) f(x)
⇒ f(t) =a 0 + a 1 t + ···+ a n−1 vt n−1 R n n
Z n 2 =(a 0a 1 ···a n−1 )
⎛
a 0 + a 1 ω + ···+ a n−1 ω n−1 ⎞ ⎛ ⎞
f(ω)
a 0 + a 1 ω 2 + ···+ a n−1 (ω 2 ) n−1
H = ⎜
⎟
⎝
⎠ = f(ω 2 )
⎜ ⎟
⎝ ⎠ =0
a 0 + a 1 ω 2r + ···+ a n−1 (ω 2r ) n−1 f(ω 2r )
f(t) C H C
⇒ f(t) =a 0 +a 1 t+···+a n−1 t n−1 C
f(ω i )=0 i =1,...,2r g(t) =[m 1 (t),...,m 2r (t)]
C = 〈g(t)〉
x 15 − 1 ∈ Z 2 [x]
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),
ω 1+x + x 4
(2 4 )
{a 0 + a 1 ω + a 2 ω 2 + a 3 ω 3 : a i ∈ Z 2 1+ω + ω 4 =0}.
ω 15 ω
m 1 (x) =1+x + x 4 ω 2 ω 4 m 1 (x)
ω 3 m 2 (x) =1+x + x 2 + x 3 + x 4
g(x) =m 1 (x)m 2 (x) =1+x 4 + x 6 + x 7 + x 8
ω ω 2 ω 3 ω 4 m 1 (x) m 2 (x) x 15 − 1
(15, 7) x 15 − 1=g(x)h(x) h(x) =1+x 4 + x 6 + x 7
⎛
⎜
⎝
0 0 0 0 0 0 0 1 1 0 1 0 0 0 1
0 0 0 0 0 0 1 1 0 1 0 0 0 1 0
0 0 0 0 0 1 1 0 1 0 0 0 1 0 0
0 0 0 0 1 1 0 1 0 0 0 1 0 0 0
0 0 0 1 1 0 1 0 0 0 1 0 0 0 0
0 0 1 1 0 1 0 0 0 1 0 0 0 0 0
0 1 1 0 1 0 0 0 1 0 0 0 0 0 0
1 1 0 1 0 0 0 1 0 0 0 0 0 0 0
⎞
.
⎟
⎠

[(3 6 ):(3 3 )]
[(128) : (16)]
[(625) : (25)]
[(p 12 ):(p 2 )]
[(p m ):(p n )] n | m
(p 30 )
α x 3 + x 2 +1 Z 2 8
x 3 + x 2 +1 Z 2 (α)
27
Q ∗
Z 2 [x]
x 5 − 1
x 6 + x 5 + x 4 + x 3 + x 2 + x +1
x 9 − 1
x 4 + x 3 + x 2 + x +1
Z 2 [x]/〈x 3 + x +1〉 ∼ = Z 2 [x]/〈x 3 + x 2 +1〉
n n =6, 7, 8, 10
〈t +1〉 R n Z n 2
7
15
p Z p (x)
p
D p (a − b) pn = a pn − b pn
a, b ∈ D
E F K E F
E = F
F ⊂ E ⊂ K K F K
E
E F F q α ∈ E
F n F (α) q n
F E
F α ∈ E E = F (α)
n n Z p [x]
Φ:(p n ) → (p n ) Φ:α ↦→ α p
n
(p n ) a p
a ∈ (p n )
E F (p n )|E| = p r |F | = p s
E ∩ F
p (p − 1)! ≡−1( p)

g(t) C R n
g(x) 1
C (n, k) n − k
R n Z n 2
C R n g(t) 〈f(t)〉 R n
〈g(t)〉 ⊂〈f(t)〉 f(x) g(x) Z 2 [x]
C = 〈g(t)〉 R n x n − 1=g(x)h(x)
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
n × k
⎛
⎞
g 0 0 ··· 0
g 1 g 0 ··· 0
G =
g n−k g n−k−1 ··· g 0
0 g n−k ··· g 1
⎜
⎝ ⎟ ⎠
0 0 ··· g n−k
H (n − k) × n
⎛
⎞
0 ··· 0 0 h k ··· h 0
0 ··· 0 h
H =
k ··· h 0 0
⎜
⎟
⎝··· ··· ··· ··· ··· ··· ··· ⎠ .
h k ··· h 0 0 0 ··· 0
G C
H C
HG =0
C R n
c(t) =c 0 + c 1 t + ···+ c n−1 t n−1
w(t) =w 0 + w 1 t + ···w n−1 t n−1 R n
a 1 ,...,a k w(t) =c(t)+e(t) e(t) =t a 1
+ t a 2
+ ···+ t a k
a i c(t)
w(t) a i w(t) w(ω i )=s i i =1,...,2r
ω n Z 2 w(t) s 1 ,...,s 2r
w(t) s i =0 i
s i = w(ω i )=e(ω i )=ω ia 1
+ ω ia 2
+ ···+ ω ia k
i =1,...,2r
s(x) =(x + ω a 1
)(x + ω a 2
) ···(x + ω a k
).

(15, 7)
a 1 a 2
s(x) =(x + ω a 1
)(x + ω a 2
)
(
s(x) =x 2 + s 1 x + s 2 1 + s )
3
.
s 1
w(t) =1+t 2 + t 4 + t 5 + t 7 + t 12 + t 13

23
x 5 − 1=0 x 6 − x 3 − 6=0
ax 5 + bx 4 + cx 3 + dx 2 + ex + f =0.
F
σ τ F στ σ −1
F
E F
E F σ : E → E
σ(α) =α α ∈ F
E F
E σ τ
E σ(α) =α τ(α) =α α ∈ F στ(α) =σ(α) =α
σ −1 (α) =α E E
F E

E F E
(E) E F
E F
G(E/F )={σ ∈ (E) :σ(α) =α α ∈ F }.
f(x) F [x] E f(x) F
f(x) G(E/F )
σ : a + bi ↦→ a − bi
σ(a) =σ(a +0i) =a − 0i = a,
G(C/R)
Q ⊂ Q( √ 5)⊂ Q( √ 3, √ 5) a, b ∈ Q( √ 5)
σ(a + b √ 3)=a − b √ 3
Q( √ 3, √ 5) Q( √ 5)
τ(a + b √ 5)=a − b √ 5
Q( √ 3, √ 5) Q( √ 3) μ = στ
√ 3 √ 5 {,σ,τ,μ} Q( √ 3, √ 5)
Q Z 2 × Z 2
σ τ μ
σ τ μ
σ σ μ τ
τ τ μ σ
μ μ τ σ
Q( √ 3, √ 5) Q {1, √ 3, √ 5, √ 15 }
|G(Q( √ 3, √ 5)/Q)| =[Q( √ 3, √ 5):Q)] = 4
E F f(x) F [x]
G(E/F ) f(x) E
f(x) =a 0 + a 1 x + a 2 x 2 + ···+ a n x n
α ∈ E f(x) σ ∈ G(E/F )
σ(α) f(x)
0=σ(0)
= σ(f(α))
= σ(a 0 + a 1 α + a 2 α 2 + ···+ a n α n )
= a 0 + a 1 σ(α)+a 2 [σ(α)] 2 + ···+ a n [σ(α)] n ;
E F α, β ∈ E
F Q( √ √ √
2)
2 − 2 Q
x 2 − 2

α β F σ :
F (α) → F (β) σ F
f(x) F [x] E
f(x) F f(x)
|G(E/F )| =[E : F ].
f(x) f(x)
0 1 E = F
k 0 ≤ k

f(x) =x 4 + x 3 + x 2 + x +1
Q f(x)
(x − 1)f(x) =x 5 − 1
f(x) ω i i =1,...,4
ω = (2π/5) + i (2π/5).
f(x) Q(ω) σ i Q(ω)
σ i (ω) =ω i i =1,...,4
G(Q(ω)/Q)
[Q(ω) :Q] =|G(Q(ω)/Q)| =4,
σ i G(Q(ω)/Q) G(Q(ω)/Q) ∼ = Z 4 ω
f(x)
F [x]
E
f(x) F [x] f(x) E
r∏
f(x) =(x − α 1 ) n 1
(x − α 2 ) n2 ···(x − α r ) nr = (x − α i ) n i
.
α i f(x) n i
f(x) ∈ F [x] n
n E f(x)
E[x] E F F
E F [x] f(x)
(f(x),f ′ (x)) = 1
f(x) F
F 0 f(x) F p f(x) ≠ g(x p )
g(x) F [x] f(x)
F =0 f ′ (x) < f(x) f(x)
(f(x),f ′ (x)) ≠1 f ′ (x)
F = p f ′ (x)
f ′ (x) p
f(x) =a 0 + a 1 x p + a 2 x 2p + ···+ a n x np
F F (α)
E F
α ∈ E E = F (α) α
Q( √ 3, √ 5)=Q( √ 3+ √ 5)
Q( 3√ 5, √ 5 i) =Q( 6√ 5 i).
i=1

E
F α ∈ E E = F (α)
F E
F (α, β)
f(x) g(x)
α β K f(x) g(x)
f(x) α = α 1 ,...,α n K g(x) β = β 1 ,...,β m K
1 E F F
a F
a ≠ α i − α
β − β j
i j j ≠1 a(β − β j ) ≠ α i − α γ = α + aβ
γ = α + aβ ≠ α i + aβ j ;
γ − aβ j ≠ α i i, j j ≠1 h(x) ∈ F (γ)[x] h(x) =f(γ − ax)
h(β) =f(α) =0 h(β j ) ≠0 j ≠1 h(x) g(x)
F (γ)[x] β F (γ)
β g(x) h(x) β ∈ F (γ) α = γ − aβ
F (γ) F (α, β) =F (γ)
G(E/F )
E F
{σ i : i ∈ I} F
F
F {σi } = {a ∈ F : σ i (a) =a σ i }
σ i (a) =a σ i (b) =b
σ i (a ± b) =σ i (a) ± σ i (b) =a ± b
σ i (ab) =σ i (a)σ i (b) =ab.
a ≠0 σ i (a −1 )=[σ i (a)] −1 = a −1 σ i (0) = 0 σ i (1) = 1 σ i
F G (F )
F
F G = {α ∈ F : σ(α) =α σ ∈ G}
F {σi } F {σ i }
G (F ) F G
σ : Q( √ 3, √ 5) → Q( √ 3, √ 5) √ 3
− √ 3 Q( √ 5) Q( √ 3, √ 5) σ

E F
E G(E/F ) = F
G = G(E/F ) F ⊂ E G ⊂ E E E G
G(E/F )=G(E/E G )
|G| =[E : E G ]=[E : F ].
[E G : F ]=1 E G = F
G E F = E G
[E : F ] ≤|G|
|G| = n n +1 α 1 ,...,α n+1 E
F a i ∈ F
a 1 α 1 + a 2 α 2 + ···+ a n+1 α n+1 =0.
σ 1 = ,σ 2 ,...,σ n G
σ 1 (α 1 )x 1 + σ 1 (α 2 )x 2 + ···+ σ 1 (α n+1 )x n+1 =0
σ 2 (α 1 )x 1 + σ 2 (α 2 )x 2 + ···+ σ 2 (α n+1 )x n+1 =0
σ n (α 1 )x 1 + σ n (α 2 )x 2 + ···+ σ n (α n+1 )x n+1 =0
x i = a i i =1, 2,...,n+1 σ 1
a 1 α 1 + a 2 α 2 + ···+ a n+1 α n+1 =0.
a i E F
a i E F α i
a 1
a 1 =1
α 2 ,...,α n+1
a 2 E F F E
G σ i G σ i (a 2 ) ≠ a 2 σ i
G
x 1 = σ i (a 1 )=1 x 2 = σ i (a 2 ) ... x n+1 = σ i (a n+1 )
x 1 =1− 1=0
x 2 = a 2 − σ i (a 2 )
x n+1 = a n+1 − σ i (a n+1 )

σ i (a 2 ) ≠ a 2
a 1 ,...,a n+1 ∈ F
E F F [x]
E E E F
F [x] E
E[x]
E F
E F
E F
F = E G G E
⇒ E F
α E E = F (α) f(x)
α F E f(x)
F E f(x)
⇒ E F
E G(E/F ) = F |G(E/F )| =[E : F ]
⇒ F = E G G E [E : F ] ≤
|G| E F E F
f(x) ∈ F [x] α E
f(x) E[x]
G f(x) E G α
α 1 = α, α 2 ,...,α n E g(x) = ∏ n
i=1 (x − α i) g(x) F
g(α) =0 σ G g(x)
σ g(x) g(x)
g(x) F g(x) ≤ f(x) f(x)
α f(x) =g(x)
K F F = K G
G K G = G(K/F )
F = K G G G(K/F )
[K : F ] ≤|G| ≤|G(K/F )| =[K : F ].
G = G(K/F )
Q( √ 3, √ 5)
Q Q
G(Q( √ 3, √ 5)/Q)

{,σ,τ,μ}
Q( √ 3, √ 5)
{,σ} {,τ} {,μ}
Q( √ 3) Q( √ 5) Q( √ 15 )
{}
Q
G(Q( √ 3, √ 5)/Q)
F
E F
G(E/F )
K ↦→ G(E/K) K E F
G(E/F )
F ⊂ K ⊂ E
[E : K] =|G(E/K)| [K : F ]=[G(E/F ):G(E/K)].
F ⊂ K ⊂ L ⊂ E {} ⊂G(E/L) ⊂ G(E/K) ⊂ G(E/F )
K F G(E/K) G(E/F )
G(K/F ) ∼ = G(E/F )/G(E/K).
G(E/K) =G(E/L) =G K L
G K = L K ↦→ G(E/K)
G G(E/F ) K G
F ⊂ K ⊂ E E K G(E/K) =G
K ↦→ G(E/K)
|G(E/K)| =[E : K]
|G(E/F )| =[G(E/F ):G(E/K)] ·|G(E/K)| =[E : F ]=[E : K][K : F ].
[K : F ]=[G(E/F ):G(E/K)]
K F σ
G(E/F ) τ G(E/K) σ −1 τσ G(E/K)
σ −1 τσ(α) =α α ∈ K f(x)
α F σ(α) f(x) K K F
τ(σ(α)) = σ(α) σ −1 τσ(α) =α
G(E/K) G(E/F )
F = K G(K/F ) τ ∈ G(E/K) σ ∈ G(E/F ) τ ∈ G(E/K)
τσ = στ α ∈ K
τ(σ(α)) = σ(τ(α)) = σ(α);

σ(α) G(E/K) σ σ K
σ K F σ(α) ∈ K α ∈ K σ ∈ G(K/F )
G(K/F ) F β K
G(K/F ) σ(β) =β σ ∈ G(E/F )
β F G(E/F )
K F
G(K/F ) ∼ = G(E/F )/G(E/K).
σ ∈ G(E/F ) σ K K σ K K
σ K ∈ G(K/F )
φ : G(E/F ) → G(K/F ) σ ↦→ σ K
φ G(E/K)
φ(στ) =(στ) K = σ K τ K = φ(σ)φ(τ).
|G(E/F )|/|G(E/K)| =[K : F ]=|G(K/F )|.
φ G(K/F ) φ
G(K/F ) ∼ = G(E/F )/G(E/K).
E
{}
L
G(E/L)
K
G(E/K)
F G(E/F )
G(E/F ) E
f(x) =x 4 − 2
Q
x 4 − 2 f(x) Q( 4√ 2,i)
f(x) (x 2 + √ 2)(x 2 − √ 2) f(x) ± 4√ 2 ± 4√ 2 i
4√ 2 Q i x 2 +1 Q( 4√ 2)
f(x) Q( 4√ 2)(i) =Q( 4√ 2,i)
[Q( 4√ 2) : Q] =4 i Q( 4√ 2) [Q( 4√ 2,i):
Q( 4√ 2)]=2 [Q( 4√ 2,i):Q] =8
{1, 4√ 2, ( 4√ 2) 2 , ( 4√ 2) 3 ,i,i 4√ 2,i( 4√ 2) 2 ,i( 4√ 2) 3 }
Q( 4√ 2,i) Q Q Q( 4√ 2,i)
G f(x) 8 σ
σ( 4√ 2) = i 4√ 2 σ(i) =i τ

τ(i) =−i G 4 2
G {,σ,σ 2 ,σ 3 ,τ,στ,σ 2 τ,σ 3 τ}
τ 2 = σ 4 = τστ = σ −1 G
D 4 G
Q( 4√ 2,i)
Q( 4√ 2) Q( 4√ 2 i) Q( √ 2,i) Q((1 + i) 4√ 2)Q((1 − i) 4√ 2)
Q( √ 2) Q(i) Q( √ 2 i)
Q
D 4
{,σ 2 ,τ,σ 2 τ}{,σ,σ 2 ,σ 3 }{,σ 2 ,στ,σ 3 τ}
{,τ} {,σ 2 τ} {,σ 2 } {,στ} {,σ 3 τ}
{}
x 4 − 2

f(x)
f(x)
n
ax 2 + bx + c =0
x = −b ± √ b 2 − 4ac
.
2a
n
E F
F = F 0 ⊂ F 1 ⊂ F 2 ⊂···⊂F r = E
i =1, 2,...,rF i = F i−1 (α i ) α n i
i
∈ F i−1 n i
f(x) F K f(x) F
F
f(x)
f(x)
x n − a
x n − 1
x n − 1 n
x n − 1 Q
1,ω,ω 2 ,...,ω n−1
( ) ( )
2π 2π
ω = + i .
n n
x n − 1 Q Q(ω)
G
G = H n ⊃ H n−1 ⊃···⊃H 1 ⊃ H 0 = {e},
H i H i+1 G {H i }
H i+1 /H i
{} ⊂A 3 ⊂ S 3 S 3 S 5

F E x n − a
F a ∈ F G(E/F )
x n − a n√ a, ω n√ a,...,ω n−1 n√ a ω n
F n ζ
x n − a x n − a ζ,ωζ,...,ω n−1 ζ E = F (ζ) G(E/F )
x n − a G(E/F )
σ τ G(E/F ) σ(ζ) =ω i ζ τ(ζ) =ω j ζF
στ(ζ) =σ(ω j ζ)=ω j σ(ζ) =ω i+j ζ = ω i τ(ζ) =τ(ω i ζ)=τσ(ζ).
στ = τσ G(E/F ) G(E/F )
F n ω
n α x n − a α ωα
x n − a ω =(ωα)/α E K = F (ω)
F ⊂ K ⊂ E K x n − 1 K F
σ G(F (ω)/F ) σ(ω)
σ(ω) =ω i i x n −1 ω τ(ω) =ω j
G(F (ω)/F )
στ(ω) =σ(ω j )=[σ(ω)] j = ω ij =[τ(ω)] i = τ(ω i )=τσ(ω).
G(F (ω)/F )
{} ⊂G(E/F (ω)) ⊂ G(E/F )
G(E/F (ω))
G(E/F )
G(E/F )/G(E/F (ω)) ∼ = G(F (ω)/F )
F
F = F 0 ⊂ F 1 ⊂ F 2 ⊂···⊂F r = E
F
F = K 0 ⊂ K 1 ⊂ K 2 ⊂···⊂K r = K
K E K i K i−1
E F
F = F 0 ⊂ F 1 ⊂ F 2 ⊂···⊂F r = E
i =1, 2,...,rF i = F i−1 (α i ) α n i
i
F
∈ F i−1 n i
F = K 0 ⊂ K 1 ⊂ K 2 ⊂···⊂K r = K
K ⊇ E K 1 x n 1
− α n 1
1
α 1 ,α 1 ω, α 1 ω 2 ,...,α 1 ω n1−1 ω n 1 F
n 1 K 1 = F (α ! )
F n 1 β x n 1
− α n 1
1

x n 1
− α n 1
1 β,ωβ,...,ω n 1−1 ω n 1
K 1 = F (ωβ) K 1 F F 1
F = K 0 ⊂ K 1 ⊂ K 2 ⊂···⊂K r = K
K i K i−1 K i ⊇ F i i =1, 2,...,r
f(x) F [x] F =0 f(x)
f(x) F
f(x) E F
F = F 0 ⊂ F 1 ⊂···⊂F n = E E
f(x) F i F i−1 G(E/F i )
G(E/F i−1 )
G(E/F )
{} ⊂G(E/F n−1 ) ⊂···⊂G(E/F 1 ) ⊂ G(E/F ).
G(E/F i−1 )/G(E/F i ) ∼ = G(F i /F i−1 ).
G(F i /F i−1 ) G(E/F )
S 5
p S p
p S p
G S p σ τ
p σ = (12) τ p τ n
p 1 ≤ n

f(x) =x 5 − 6x 3 − 27x − 3
f(x) =x 5 − 6x 3 − 27x − 3 ∈ Q[x]
f(x) Q S 5 f(x)
f(x) f ′ (x) =5x 4 −18x 2 −
27 f ′ (x) =0 f ′ (x)
√
6 √ 6+9
x = ± .
5
f(x)
f(x) −3 −2 −2 0 0
4 f(x)
f(x) K f(x) f(x)
K K Q
f(x) G(K/Q) S 5 f
σ ∈ G(K/Q) σ(a) =b a b
f(x) C a + bi ↦→ a − bi
G(K/Q) α
f(x) [Q(α) :Q] =5 Q(α)
K [K : Q] [K : Q] =|G(K/Q)|
G(K/Q) ⊂ S 5 G(K/Q) 5 S 5
5 G(K/Q)

S 5 S 5 f(x)
C[x] C
E
α ∈ E E = C(α) α f(x) C[x]
L f(x) C E
L C
L C L f(x)(x 2 +1)
R L R K
G G(L/R) L ⊃ K ⊃ R |G(L/K)| =[L : K] [L : R] =[L :
K][K : R] [K : R] K = R(β) β
f(x) K = R
G(L/R) G(L/C) 2
L ≠ C |G(L/C)| ≥2
G G(L/C)
E G [E : C] =2 γ ∈ E
x 2 + bx + c C[x] (−b ± √ b 2 − 4c )/2
C b 2 − 4c C L = C
Q

G(Q( √ 30 )/Q)
G(Q( 4√ 5)/Q)
G(Q( √ 2, √ 3, √ 5)/Q)
G(Q( √ 2, 3√ 2,i)/Q)
G(Q( √ 6,i)/Q)
x 3 +2x 2 − x − 2 Q
x 4 +2x 2 +1 Q
x 4 + x 2 +1 Z 3
x 3 + x 2 +1 Z 2
(729) (9)
Q[x]
x 5 − 12x 2 +2
x 5 − 4x 4 +2x +2
x 3 − 5
x 4 − x 2 − 6
x 5 +1
(x 2 − 2)(x 2 +2)
x 8 − 1
x 8 +1
x 4 − 3x 2 − 10
Q[x]
x 4 − 1
x 4 − 8x 2 +15
x 4 − 2x 2 − 15
x 3 − 2
Z 2
S 3
Z 3
F ⊂ K ⊂ E E F E
K
G n |G| n!
F ⊂ E f(x) F f(x) E
f(x) Q[x] 7
p f(x) ∈ Q[x] p
S p p p ≥ 5
p
p Z p (t) Z p
f(x) =x p − t Z p (t)[x] f(x)
E F K L
σ ∈ G(E/F ) σ(K) =L K L
K L G(E/K) G(E/L)
G(E/F )
σ ∈ (R) a σ(a) > 0

K x 3 + x 2 +1∈ Z 2 [x] K
F (F ) ≠2 f(x) =
ax 2 + bx + c F ( √ α ) α = b 2 − 4ac
K F E F
K [E : F ]=2 E F [x]
Φ p (x) = xp − 1
x − 1 = xp−1 + x p−2 + ···+ x +1
Q p ω Φ p (x)
Q(ω)
ω, ω 2 ,...,ω p−1 Φ p (x)
Φ p (x)
G(Q(ω)/Q) p − 1
G(Q(ω)/Q) Q
F E
F G(E/F ) F ⊂ K ⊂ L ⊂ E
{} ⊂G(E/L) ⊂ G(E/K) ⊂ G(E/F )
F f(x) ∈ F [x]
n E f(x) α 1 ,...,α n f(x) E
Δ= ∏ i

A

B
A ∩ B = {2} B ∩ C = {5}
A × B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c, 1), (c, 2), (c, 3)}
A × D = ∅
x ∈ A ∪ (B ∩ C) x ∈ A x ∈ B ∩ C x ∈ A ∪ B
A ∪ C x ∈ (A ∪ B) ∩ (A ∪ C) A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C)
x ∈ (A ∪ B) ∩ (A ∪ C) x ∈ A ∪ B A ∪ C x ∈ A x
B C x ∈ A ∪ (B ∩ C) (A ∪ B) ∩ (A ∪ C) ⊂ A ∪ (B ∩ C)
A ∪ (B ∩ C) =(A ∪ B) ∩ (A ∪ C)
(A ∩ B) ∪ (A \ B) ∪ (B \ A) =(A ∩ B) ∪ (A ∩ B ′ ) ∪ (B ∩ A ′ )=[A ∩ (B ∪
B ′ )] ∪ (B ∩ A ′ )=A ∪ (B ∩ A ′ )=(A ∪ B) ∩ (A ∪ A ′ )=A ∪ B
A \ (B ∪ C) =A ∩ (B ∪ C) ′ =(A ∩ A) ∩ (B ′ ∩ C ′ )=(A ∩ B ′ ) ∩ (A ∩ C ′ )=
(A \ B) ∩ (A \ C)
f(2/3)
f(1/2) = 3/4 f(2/4) = 3/8
f f(R) ={x ∈ R : x>0} f
f(R) ={x : −1 ≤ x ≤ 1}
f(n) =n +1
x, y ∈ A g(f(x)) = (g◦f)(x) =(g◦f)(y) =g(f(y))
f(x) =f(y) x = y g ◦ f c ∈ C c =(g ◦ f)(x) =g(f(x))
x ∈ A f(x) ∈ B g
f −1 (x) =(x +1)/(x − 1)
y ∈ f(A 1 ∪ A 2 ) x ∈ A 1 ∪ A 2
f(x) =y y ∈ f(A 1 ) f(A 2 ) y ∈ f(A 1 ) ∪ f(A 2 )
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 )
x A 1 A 2 f(x) =y x ∈ A 1 ∪ A 2
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 )
0
X = N ∪{ √ 2 } x ∼ y x + y ∈ N

S(1) : [1(1 + 1)(2(1) + 1)]/6 = 1 = 1 2
S(k) :1 2 +2 2 + ···+ k 2 =[k(k + 1)(2k +1)]/6
1 2 +2 2 + ···+ k 2 +(k +1) 2 =[k(k + 1)(2k +1)]/6+(k +1) 2
=[(k + 1)((k + 1) + 1)(2(k + 1) + 1)]/6,
S(k +1) S(n) n
S(4):4!=24> 16 = 2 4 S(k) :k! > 2 k
(k +1)!=k!(k +1)> 2 k · 2=2 k+1 S(k +1) S(n)
n
S(0) : (1 + x) 0 − 1=0≥ 0=0· x
S(k) :(1+x) k − 1 ≥ kx
(1 + x) k+1 − 1=(1+x)(1 + x) k − 1
=(1+x) k + x(1 + x) k − 1
≥ kx + x(1 + x) k
≥ kx + x
=(k +1)x,
S(k +1) S(n) n
f 1 =1 f 2 =1 f n+2 = f n+1 + f n
(a, b) =1 r s ar + bs =1
acr + bcs = c
2 3 6n +16n +5
6k +5
3+7Z = {...,−4, 3, 10,...} 18 + 26Z 5+6Z
· 1 5 7 11
1 1 5 7 11
5 5 1 11 7
7 7 11 1 5
11 11 7 5 1

( 1 2 ···
σ =
a 1 a 2 ···
) n
a n
S n a i n a 1 n − 1
a 2 ... a n−1 a n
σ n(n − 1) ···2 · 1=n!
(aba −1 ) n =(aba −1 )(aba −1 ) ···(aba −1 )
= ab(aa −1 )b(aa −1 )b ···b(aa −1 )ba −1
= ab n a −1 .
abab =(ab) 2 = e = a 2 b 2 = aabb ba = ab
H 1 = {} H 2 = {,ρ 1 ,ρ 2 } H 3 = {,μ 1 } H 4 = {,μ 2 } H 5 =
{,μ 3 } S 3
G 1=1+0 √ 2 (a + b √ 2)(c + d √ 2) = (ac +
2bd)+(ad + bc) √ 2 G (a + b √ 2) −1 = a/(a 2 − 2b 2 ) −
b √ 2/(a 2 − 2b 2 )
S 3
ba = a 4 b = a 3 ab = ab
12 10
7Z = {...,−7, 0, 7, 14,...} {0, 3, 6, 9, 12, 15, 18, 21} {0} {0, 6}
{0, 4, 8} {0, 3, 6, 9} {0, 2, 4, 6, 8, 10} {1, 3, 7, 9} {1, −1,i,−i}
( ) ( )
1 0 −1 0
,
,
0 1 0 −1
( ) 0 −1
,
1 0
( ) 0 1
.
−1 0
( ) 1 0
,
0 1
( ) 1 −1
,
1 0
( ) −1 1
,
−1 0
( ) 0 1
,
−1 1
( ) ( )
0 −1 −1 0
,
.
1 −1 0 −1
0 1, −1
1, 2, 3, 4, 6, 8, 12, 24
−3+3i 43 − 18i i
√ 3+i −3

√ 2 (7π/4) 2 √ 2 (π/4) 3 (3π/2)
(1 − i)/2 16(i − √ 3) −1/4
292 1523
|〈g〉∩〈h〉| =1
g, h ∈ G
m n (g −1 ) m = e (gh) mn = e
G G
g G g G
〈g〉 G
(12453) (13)(25)
(135)(24) (14)(23) (1324) (134)(25) (17352)
(16)(15)(13)(14) (16)(14)(12)
(a 1 ,a 2 ,...,a n ) −1 =(a 1 ,a n ,a n−1 ,...,a 2 )
{(13), (13)(24), (132), (134), (1324), (1342)}
(12345)(678)
(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 )
A 5
(123)(12) (12)(123)
(ab)(bc) (ab)(cd)
στσ −1 (σ(a i )) = σ(a i+1 )
g h G
60
〈8〉 1+〈8〉 2+〈8〉 3+〈8〉 4+〈8〉 5+〈8〉 6+〈8〉 7+〈8〉
3Z 1+3Z 2+3Z
4 φ(15) ≡ 4 8 ≡ 1( 15)
g 1 ∈ gH g 1 ∈ Hg gH ⊂ Hg
g(H ∩ K) =gH ∩ gK
(m, n) =1 φ(mn) =φ(m)φ(n)

26! − 1
2791 11213525032442
31 14
n =11· 41 n = 8779 · 4327
(0000) /∈ C
2 2
3 4
d =2 d =1
(00000), (00101), (10011), (10110)
⎛ ⎞
0 1
0 0
G =
1 0
⎜ ⎟
⎝0 1⎠
1 1
(000000), (010111), (101101), (111010)
⎛ ⎞
1 0
0 1
1 0
G =
1 1
⎜ ⎟
⎝0 1⎠
1 1
⎛ ⎞
1
1
G =
0
.
⎜ ⎟
⎝0⎠
1

⎛ ⎞
1 0
0 1
G = ⎜ ⎟
⎝1 1⎠ .
1 0
C (10000) + C (01000) + C (00100) + C (00010) + C (11000) + C
(01100) + C (01010) + C C
∈ C
↦→ +
20
Z
φ : C ∗ → GL 2 (R)
( ) a b
φ(a + bi) = .
−b a
Z n n k ↦→ (2kπ/n)
Q
12 5
a G φ : G → H
φ(a) H
Z 6 Z 6
φ g 1 = h 1 k 1 g 2 = h 2 k 2
φ(g 1 )=φ(g 2 )

A 4 (12)A 4
A 4 A 4 (12)A 4
(12)A 4 (12)A 4 A 4
D 4 S 4
a ∈ G G aH G/H
g ∈ G i g : G → G i g : x ↦→ gxg −1
G i g (H)
〈g〉 G y
G x ∈ C(g) yxy −1 C(g) (yxy −1 )g = g(yxy −1 )
g ∈ G h ∈ G ′ h = aba −1 b −1
ghg −1 = gaba −1 b −1 g −1
=(gag −1 )(gbg −1 )(ga −1 g −1 )(gb −1 g −1 )
=(gag −1 )(gbg −1 )(gag −1 ) −1 (gbg −1 ) −1 .
h = h 1 ···h n h i = a i b i a −1
i
b −1
i
ghg −1
ghg −1 = gh 1 ···h n g −1 =(gh 1 g −1 )(gh 2 g −1 ) ···(gh n g −1 )
{1}
φ(m + n) =7(m + n) =7m +7n = φ(m)+φ(n) φ
φ : Z 24 → Z 18 φ
Z 24 φ Z 18
a, b ∈ G φ(a)φ(b) =φ(ab) =φ(ba) =φ(b)φ(a)
1 [
‖ + ‖ 2 + ‖‖ 2 −‖‖ 2] = 1 [
〈x + y, x + y〉−‖‖ 2 −‖‖ 2]
2
2
= 1 [
‖‖ 2 +2〈x, y〉 + ‖‖ 2 −‖‖ 2 −‖‖ 2]
2
= 〈, 〉.

SO(2) O(3)
〈, 〉 = 〈, 〉
( ) 5 2
.
2 1
: O(n) → R ∗ SO(n)
p6m
{0} ⊂〈6〉 ⊂〈3〉 ⊂Z 12 {(1)}×{0} ⊂{(1), (123), (132)}×{0} ⊂
S 3 ×{0} ⊂S 3 ×〈2〉 ⊂S 3 × Z 4
N G/N
N = N n ⊃ N n−1 ⊃···⊃N 1 ⊃ N 0 = {e}
G/N = G n /N ⊃ G n−1 /N ⊃···G 1 /N ⊃ G 0 /N = {N}.
D n 2
G/G ′
0 R 2 \{0} X = {1, 2, 3, 4}
X (1) = {1, 2, 3} X (12) = {3} X (13) = {2} X (23) = {1} X (123) =
X (132) = ∅ G 1 = {(1), (23)} G 2 = {(1), (13)} G 3 = {(1), (12)}
O 1 = O 2 = O 3 = {1, 2, 3}
S 4
O (1) = {(1)},
O (12) = {(12), (13), (14), (23), (24), (34)},
O (12)(34) = {(12)(34), (13)(24), (14)(23)},
O (123) = {(123), (132), (124), (142), (134), (143), (234), (243)},
O (1234) = {(1234), (1243), (1324), (1342), (1423), (1432)}.
1+3+6+6+8=24
(3 4 +3 1 +3 2 +3 1 +3 2 +3 2 +3 3 +3 3 )/8 = 21

S 4
(abcd)
(ab)(cd)
(ab)(cd)(ef)
(abc)(def)
(1 · 2 6 +3· 2 4 +4· 2 3 +2· 2 2 +2· 2 1 )/12 = 13
(1 · 2 8 +3· 2 6 +2· 2 4 )/6 = 80
x ∈ gC(a)g −1 g −1 xg ∈ C(a)
|G| =18=2· 3 2 2 2
3 9
3 S 4 P 1 = {(1), (123), (132)} P 2 =
{(1), (124), (142)} P 3 = {(1), (134), (143)} P 4 = {(1), (234), (243)}
|G| =96=2 5 · 3 G 2
2 H K |H ∩ K| ≥16 HK
(32 · 32)/8 = 128 H ∩ K
H K 2
G p p 2
q q 2
G G |G| =3· 5 · 17
N(H) G
H G N(H)g ↦→ g −1 Hg
aG ′ ,bG ′ ∈ G/G ′ (aG ′ )(bG ′ )=abG ′ = ab(b −1 a −1 ba)G ′ =
(abb −1 a −1 )baG ′ = baG ′
7Z Q( √ 2) R
{1, 3, 7, 9} {1, 2, 3, 4, 5, 6}
{( ) 1 0
,
0 1
( ) 1 1
,
0 1
( ) 1 0
,
1 1
( ) 0 1
,
1 0
( ) 1 1
,
1 0
( ) } 0 1
, .
1 1
{0} {0, 9} {0, 6, 12} {0, 3, 6, 9, 12, 15} {0, 2, 4, 6, 8, 10, 12, 14, 16}
φ : C → R φ(i) =a

φ( √ 2)=a
φ : Q( √ 2) → Q( √ 3)
x ≡ 17 ( 55) x ≡ 214 ( 2772)
I ≠ {0} 1 ∈ I
φ(a)φ(b) =φ(ab) =φ(ba) =φ(b)φ(a)
a ∈ R a ≠0 a R
b ∈ R ab =1
(a + b) 2 (−ab) 2
a/b, c/d ∈ Z (p) a/b + c/d =(ad + bc)/bd (a/b) · (c/d) =
(ac)/(bd) Z (p) (bd, p) =1
x 2 = x x ≠0 R x =1
M 2 (R)
9x 2 +2x +5 8x 4 +7x 3 +2x 2 +7x
5x 3 +6x 2 − 3x +4 = (5x 2 +2x +1)(x − 2) + 6 4x 5 − x 3 + x 2 +4 =
(4x 2 +4)(x 3 +3)+4x 2 +2
Z 12 3 4
(2x +1)
x 2 + x +8=(x +2)(x +9)
Z
φ : R → S φ : R[x] → S[x]
φ(a 0 + a 1 x + ···+ a n x n )=φ(a 0 )+φ(a 1 )x + ···+ φ(a n )x n
Φ n (x) = xn − 1
x − 1 = xn−1 + x n−2 + ···+ x +1
Φ p (x) Q
p
F [x]
z −1 = 1/(a + b √ 3 i) =(a − b √ 3 i)/(a 2 +3b 2 ) Z[ √ 3 i]
a 2 +3b 2 =1 a = ±1,b=0
5=−i(1 + 2i)(2 + i) 6+8i = −i(1 + i) 2 (2 + i) 2
z = a + bi w = c + di ≠0 Z[i] z/w ∈ Q(i)

a = ub u ν(b) ≤ ν(ub) ≤ ν(a)
ν(a) ≤ ν(b)
(a ∨ b ∨ a ′ ) ∧ a
a
b
a
a ′
a ∨ (a ∧ b)
a
b
a
a ′ ∧ [(a ∧ b ′ ) ∨ b] =a ∧ (a ∨ b)

I,J R I + J = {r + s :
r ∈ I s ∈ J} R I J r 1 ,r 2 ∈ I
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
a(r 1 + s 1 )=ar 1 + as 1 ∈ I + J I + J R
(⇒) a = b ⇒ (a ∧ b ′ ) ∨ (a ′ ∧ b) =(a ∧ a ′ ) ∨ (a ′ ∧ a) =O ∨ O = O
(⇐) (a ∧ b ′ ) ∨ (a ′ ∧ b) =O ⇒ a ∨ b =(a ∨ a) ∨ b = a ∨ (a ∨ b) =a ∨ [I ∧ (a ∨ b)] =
a ∨ [(a ∨ a ′ ) ∧ (a ∨ b)] = [a ∨ (a ∧ b ′ )] ∨ [a ∨ (a ′ ∧ b)] = a ∨ [(a ∧ b ′ ) ∨ (a ′ ∧ b)] = a ∨ 0=a
a ∨ b = b
Q( √ 2, √ 3) {1, √ 2, √ 3, √ 6 } Q
{1,x,x 2 ,...,x n−1 } P n
2 {(1, 0, −3), (0, 1, 2)}
0=α0 =α(−v + v) =α(−v)+αv −αv = α(−v)
v 0 =0,v 1 ,...,v n ∈ V α 0 ≠0,α 1 ,...,α n ∈ F α 0 v 0 +
···+ α n v n =0
u, v ∈ (T ) α ∈ F
T (u + v) =T (u)+T (v) =0
T (αv) =αT (v) =α0 =0.
u + v, αv ∈ (T ) (T ) V
T (u) =T (v) T (u − v) =T (u) − T (v) =0
u − v =0 u = v
u, u ′ ∈ U v, v ′ ∈ V
(u + v)+(u ′ + v ′ )=(u + u ′ )+(v + v ′ ) ∈ U + V
α(u + v) =αu + αv ∈ U + V.
x 4 − (2/3)x 2 − 62/9 x 4 − 2x 2 +25
{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 }
Q( √ 3, √ 7)
Z 2 [x]/〈x 3 + x +1〉 α 1+α
α 2 1+α 2 α + α 2 1+α + α 2 α 3 + α +1=0
E F K E
α ∈ K α F α

E p(x) =β 0 + β 1 x + ···+ β n x n
E[x] α F (β 0 ,...,β n )
{1, √ 3, √ 7, √ 21 } Q( √ 3, √ 7) Q Q( √ 3, √ 7)⊃
Q( √ 3+ √ 7) [Q( √ 3, √ 7) : Q] =4 [Q( √ 3+ √ 7) : Q] =2
√ 3+ √ 7 Q( √ 3, √ 7)=Q( √ 3+ √ 7)
β ∈ F (α) F β = p(α)/q(α) p q
α q(α) ≠0 F β F
f(x) ∈ F [x] f(β) =0 f(x) =a 0 + a 1 x + ···+ a n x n
( ) ( ) ( ) p(α)
p(α)
p(α) n
0=f(β) =f = a 0 + a 1 + ···+ a n .
q(α)
q(α)
q(α)
q(α) n F [x] α
Z 2 (α) x 3 + x 2 +1
α
p(x) Z 3 [x] 3
Z 3 [x]/〈p(x)〉 27
x 5 − 1=(x +1)(x 4 + x 3 + x 2 + x +1) x 9 − 1=(x +1)(x 2 + x +
1)(x 6 + x 3 +1)
x 7 − 1=(x +1)(x 3 + x +1)(x 3 + x 2 +1)
p(x) ∈ F [x] p(x) ∈ E[x]
α F n
β ∈ F (α) β = a 0 + a 1 α + ···+ a n−1 α n−1 a i ∈ F q n
n (a 0 ,a 1 ,...,a n−1 )
x p−1 − 1 Z p
Z 2 Z 2 × Z 2 × Z 2
Q x 3 +2x 2 − x − 2=(x − 1)(x +1)(x +2)
Z 3 x 4 + x 2 +1=(x +1) 2 (x +2) 2
[(729) : (9)] = [(729) : (3)]/[(9) : (3)] = 6/2 = 3,

G((729)/ (9)) ∼ = Z 3 G((729)/ (9)) σ σ 3 6(α) =
α 36 = α 729 α ∈ (729)
S 5 S 3
Q(i)
E F [x]
[E : F ] 6 3 G(E/F ) S 3
3 Z 3 S 3
G S n
ω, ω 2 ,...,ω p−1 ω ≠1 ω i Φ p
Φ p (ω i )
ω ω, ω 2 ,...,ω p−1 φ i : Q(ω) → Q(ω i )
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 ,
a i ∈ Q φ i φ 2
G(Q(ω)/Q)
{ω, ω 2 ,...,ω p−1 } Q(ω) Q
ω, ω 2 ,...,ω p−1 G(Q(ω)/Q)

C
a ∈ A a A
N
Z
Q
R
C
A ⊂ B A B
∅
A ∪ B A B
A ∩ B A B
A ′ A
A \ B A B
A × B A B
A n A ×···×A n
id
f −1 f
a ≡ b ( n) a b n
n! n
( n
k)
n!/(k!(n − k)!)
a | b a b
(a, b) a b
P(X) X
(m, n) m n
Z n n
U(n) Z n
M n (R) n × n R
A A
GL n (R)
Q 8
C ∗
|G|

R ∗
Q ∗
SL n (R)
Z(G)
〈a〉 a
|a| a
θ θ + i θ
T
S n n
(a 1 ,a 2 ,...,a k ) k
A n n
D n
[G : H] H G
L H H G
R H H G
a ∤ b a b
d(, )
d
w()
M m×n ( 2 ) m × n Z 2
(H) H
δ ij
G ∼ = H G H
(G) G
i g i g (x) =gxg −1
(G) G
ρ g
G/N G N
G ′ G
φ φ
(a ij )
O(n)
‖‖
SO(n)
E(n)
O x x
X g g
G x x
N(H) H
H
Z[i]
R R
Z (p) p
f(x)

R[x] R
R[x 1 ,x 2 ,...,x n ] n
φ α α
Q(x) Q
ν(a) a
F (x) x
F (x 1 ,...,x n ) x 1 ,...,x n
a ≼ b a b
a ∨ b a b
a ∧ b a b
I
O
a ′ a
V V
U ⊕ V U V
(V,W) U V
V ∗ V
F (α 1 ,...,α n ) F α 1 ,...,α n
[E : F ] E F
(p n ) p n
F ∗ F
G(E/F ) E F
F {σi } σ i
F G G
Δ 2

G
G
n
n

φ

p
p

n
n
p
p
p