Commutative algebra - Department of Mathematical Sciences - old ...
Commutative algebra - Department of Mathematical Sciences - old ...
Commutative algebra - Department of Mathematical Sciences - old ...
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18 1. A DICTIONARY ON RINGS AND IDEALS<br />
(2) If S is a finite type ring over R, then S is a factor ring <strong>of</strong> a polynomial ring<br />
in finitely many variables over R.<br />
1.6.13. Exercise. (1) Let K be a field. Show that there are infinitely many prime ideals<br />
in K[X].<br />
(2) What are the units in the ring Z[X]/(1 − 2X)?<br />
(3) Determine the prime ideals in Q[X]/(X − X 2 ).<br />
(4) Show that the ring Z[X] is not a principal ideal domain.<br />
(5) Show that the ring Q[X, Y ] is not a principal ideal domain.<br />
1.7. Roots<br />
1.7.1. Definition. Let φ : R → S be a ring homomorphism and f ∈ R[X] a<br />
polynomial. An element b ∈ S is a root <strong>of</strong> f (in S) if the evaluation f(b) = 0.<br />
1.7.2. Proposition. Let R be a domain. An element a ∈ R is a root <strong>of</strong> the polynomial<br />
f ∈ R[X] if and only if there is a q ∈ R[X] such that<br />
f = q(X − a)<br />
Pro<strong>of</strong>. By 1.6.6 f = q(X −a)+r. It follows that a is a root if and only if r = 0.<br />
1.7.3. Corollary. Let R is a domain. There are at most deg(f) roots in a nonzero<br />
polynomial f ∈ R[X].<br />
1.7.4. Definition. The multiplicity <strong>of</strong> a root a <strong>of</strong> a nonzero polynomial f ∈ R[X]<br />
is highest m such that<br />
f = q(X − a) m<br />
A root <strong>of</strong> multiplicity m = 1 is a simple root.<br />
1.7.5. Corollary. Let R is a domain. If m1, . . . , mk are the multiplicities <strong>of</strong> the<br />
roots <strong>of</strong> a nonzero polynomial f ∈ R[X], then m1 + · · · + mk ≤ deg(f).<br />
1.7.6. Definition. The derivative <strong>of</strong> a polynomial f = anX n ∈ R[X] is<br />
1.7.7. Lemma. The derivative satisfies<br />
(1) (f + g) ′ = f ′ + g ′ .<br />
(2) (fg) ′ = f ′ g + fg ′<br />
(3) If f is constant, then f ′ = 0.<br />
f ′ = nanX n−1<br />
1.7.8. Proposition. Let R is a domain. An element a ∈ R is a root <strong>of</strong> multiplicity<br />
m > 1 <strong>of</strong> a nonzero f ∈ R[X] if and only if a is a root <strong>of</strong> f and f ′ .<br />
Pro<strong>of</strong>. By 1.6.6 f = q(X − a) 2 + cX + d and by 1.7.7 f ′ = q ′ (X − a) 2 + 2q(X −<br />
a) + c. I follows that a is a root <strong>of</strong> multiplicity m > 1 if and only if c = d = 0.<br />
1.7.9. Exercise. (1) Let a1, . . . ak be roots with multiplicities m1, . . . , mk in a polynomial<br />
f. Show that m1 + · · · + mk ≤ deg(f).<br />
(2) Let K be a field and let a1, . . . , an ∈ K. Show that the ideal (X1 −a1, . . . , Xn −an)<br />
is maximal in K[X1, . . . , Xn].<br />
(3) Let the characteristic char(R) = n > 0. What is (X n ) ′ in R[X].