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18 1. A DICTIONARY ON RINGS AND IDEALS (2) If S is a finite type ring over R, then S is a factor ring of a polynomial ring in finitely many variables over R. 1.6.13. Exercise. (1) Let K be a field. Show that there are infinitely many prime ideals in K[X]. (2) What are the units in the ring Z[X]/(1 − 2X)? (3) Determine the prime ideals in Q[X]/(X − X 2 ). (4) Show that the ring Z[X] is not a principal ideal domain. (5) Show that the ring Q[X, Y ] is not a principal ideal domain. 1.7. Roots 1.7.1. Definition. Let φ : R → S be a ring homomorphism and f ∈ R[X] a polynomial. An element b ∈ S is a root of f (in S) if the evaluation f(b) = 0. 1.7.2. Proposition. Let R be a domain. An element a ∈ R is a root of the polynomial f ∈ R[X] if and only if there is a q ∈ R[X] such that f = q(X − a) Proof. By 1.6.6 f = q(X −a)+r. It follows that a is a root if and only if r = 0. 1.7.3. Corollary. Let R is a domain. There are at most deg(f) roots in a nonzero polynomial f ∈ R[X]. 1.7.4. Definition. The multiplicity of a root a of a nonzero polynomial f ∈ R[X] is highest m such that f = q(X − a) m A root of multiplicity m = 1 is a simple root. 1.7.5. Corollary. Let R is a domain. If m1, . . . , mk are the multiplicities of the roots of a nonzero polynomial f ∈ R[X], then m1 + · · · + mk ≤ deg(f). 1.7.6. Definition. The derivative of a polynomial f = anX n ∈ R[X] is 1.7.7. Lemma. The derivative satisfies (1) (f + g) ′ = f ′ + g ′ . (2) (fg) ′ = f ′ g + fg ′ (3) If f is constant, then f ′ = 0. f ′ = nanX n−1 1.7.8. Proposition. Let R is a domain. An element a ∈ R is a root of multiplicity m > 1 of a nonzero f ∈ R[X] if and only if a is a root of f and f ′ . Proof. By 1.6.6 f = q(X − a) 2 + cX + d and by 1.7.7 f ′ = q ′ (X − a) 2 + 2q(X − a) + c. I follows that a is a root of multiplicity m > 1 if and only if c = d = 0. 1.7.9. Exercise. (1) Let a1, . . . ak be roots with multiplicities m1, . . . , mk in a polynomial f. Show that m1 + · · · + mk ≤ deg(f). (2) Let K be a field and let a1, . . . , an ∈ K. Show that the ideal (X1 −a1, . . . , Xn −an) is maximal in K[X1, . . . , Xn]. (3) Let the characteristic char(R) = n > 0. What is (X n ) ′ in R[X].
1.8. FIELDS 19 1.8. Fields 1.8.1. Definition. Let p ∈ Z be a prime number. The factor ring Fp = Z/(p) is a field with p elements. Together with Q they constitute the prime fields. 1.8.2. Proposition. Let K be a field then the polynomial ring K[X] is a principal ideal domain. Proof. Let d = 0 be a polynomial of lowest degree in an ideal I. Given f ∈ I then by 1.6.6 f = qd + r with r = f − qd ∈ I. By degree considerations r = 0 and I = (d). 1.8.3. Corollary. Let K be a field then the polynomial ring K[X] is a unique factorization domain. Proof. Follows from 1.5.6. 1.8.4. Definition. A subfield is a subring, which is a field. A field extension is the inclusion of a subfield K ⊂ L in a field. A finite field extension K ⊂ L is an extension, where L is finite dimensional as a vector space over K. 1.8.5. Example. (1) Let K be a field and f an irreducible polynomial in K[X]. Then K ⊂ K[X]/(f) is a finite field extension. (2) Let K ⊂ R ⊂ L be a subring in a finite field extension. Then R is a field. Namely multiplication on R with a nonzero element of R is a K-linear map on the finite dimensional K-vector space R and therefore an isomorphism. 1.8.6. Proposition. (1) Let K be a field and f a polynomial in K[X]. Then there is a finite field extension K ⊂ L such that f factors in linear factors in L[X]. (2) If K ⊂ L1 and K ⊂ L2 are finite field extensions then there is a finite field extension K ⊂ L such that L1 ∪ L2 ⊂ L. Proof. (1) Assume f irreducible. In L = K[X]/(f) the class X + (f) is a root of f. In general proceed adjoining roots of irreducible factors of f. (2) An element x in a finite field extension K ⊂ K ′ is the root of the irreducible monic polynomial f generating the kernel of the evaluation homomorphism K[X] → K ′ , X ↦→ x. Now proceed by (1) adjoining elements in L2 to L1. 1.8.7. Proposition. Let p ∈ Z be a prime number. For any power q = p n there is a field Fq with q elements, unique up to isomorphism. Proof. Let Fp ⊂ K be a field extension, where X q − X factors into linear factors, 1.8.6 (1). The subset of roots is the set of elements fixed under n-times the Frobenius and therefore a subring being a subfield by 1.8.5 (2). The derivative (X q − X) ′ = −1 so by 1.7.8 there are q elements in this subfield. Uniqueness follows from 1.8.6 (2). 1.8.8. Exercise. (1) Show that the ring R[X]/(X 2 + 1) is isomorphic to the field of complex numbers. (2) Show that the ring F2[X]/(X 2 + X + 1) is a field with 4 elements. (3) Show that the ring F2[X]/(X 3 + X + 1) is a field with 8 elements. (4) Let K ⊂ L be a field extension of fields of characteristic 0 and let a ∈ L be a root of an irreducible polynomial f ∈ K[X]. Show that a is a simple root. (5) Let p be a prime number. Show that Fp is the only ring with p elements. (6) Let p be a prime number. Show that a ring with p 2 elements is isomorphic to one of four non isomorphic Z/(p 2 ), Fp × Fp, Fp[X]/(X 2 ), F p 2.
Page 1: ELEMENTARY COMMUTATIVE ALGEBRA LECT
Page 5 and 6: Contents Prerequisites 7 1. A dicti
Page 7: Prerequisites The basic notions fro
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68 5. LOCALIZATION (2) Any local ri
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70 5. LOCALIZATION 5.4. Exactness a
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72 5. LOCALIZATION (1) φ is flat.
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74 6. FINITE MODULES 6.1.7. Corolla
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76 6. FINITE MODULES (4) Let A be a
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78 6. FINITE MODULES Let the n × n
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80 6. FINITE MODULES 6.5. Finite Pr
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82 6. FINITE MODULES 6.5.9. Corolla
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84 6. FINITE MODULES (1) E is an in
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86 7. MODULES OF FINITE LENGTH 7.2.
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88 7. MODULES OF FINITE LENGTH Proo
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90 7. MODULES OF FINITE LENGTH (5)
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92 7. MODULES OF FINITE LENGTH 7.5.
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94 8. NOETHERIAN MODULES Proof. Use
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96 8. NOETHERIAN MODULES 8.3.2. Cor
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98 8. NOETHERIAN MODULES 8.5. Fract
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9 Primary decomposition 9.1. Suppor
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9.2. ASS OF MODULES 103 Proof. The
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9.2. ASS OF MODULES 105 9.2.12. Def
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9.4. DECOMPOSITION OF MODULES 107 9
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9.5. DECOMPOSITION OF IDEALS 109 9.
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10 Dedekind rings 10.1. Principal i
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10.3. DEDEKIND DOMAINS 113 10.2.4.
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10.3. DEDEKIND DOMAINS 115 10.3.10.
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118 BIBLIOGRAPHY
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120 INDEX finite type rings, 95 fin
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