A Brief Introduction to Classical and Adelic Algebraic ... - William Stein

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172 CHAPTER 21. IDELES AND IDEALS Proof. We first prove that the map I1 K → FK is surjective. Let ∞ be an archimedean valuation on K. If v is a non-archimedean valuation, let x ∈ I1 K be a 1-idele such that xw = 1 at ever valuation w except v and ∞. At v, choose xv = π to be a generator for the maximal ideal of Ov, and choose x∞ to be such that |x∞| ∞ = 1/ |xv| v . Then x ∈ IK and w |xw| w = 1, so x ∈ I1 K . Also x maps to v ∈ FK. Thus the group of ideal classes is the continuous image of the compact group I1 K /K∗ (see Theorem 21.1.12), hence compact. But a compact discrete group is finite. 21.2.1 The Function Field Case When K is a finite separable extension of F(t), we define the divisor group DK of K to be the free abelian group on all the valuations v. For each v the number of elements of the residue class field Fv = Ov/℘v of v is a power, say q nv , of the number q of elements in Fv. We call nv the degree of v, and similarly define nvdv to be the degree of the divisor nv · v. The divisors of degree 0 form a group D 0 K . As before, the principal divisor attached to a ∈ K ∗ is ordv(a) · v ∈ DK. The following theorem is proved in the same way as Theorem 21.2.2. Theorem 21.2.3. The quotient of D0 K modulo the principal divisors is a finite group. 21.2.2 Jacobians of Curves For those familiar with algebraic geometry and algebraic curves, one can prove Theorem 21.2.3 from an alternative point of view. There is a bijection between nonsingular geometrically irreducible projective curves over F and function fields K over F (which we assume are finite separable extensions of F(t) such that F∩K = F). Let X be the curve corresponding to K. The group D 0 K is in bijection with the divisors of degree 0 on X, a group typically denoted Div 0 (X). The quotient of Div 0 (X) by principal divisors is denoted Pic 0 (X). The Jacobian of X is an abelian variety J = Jac(X) over the finite field F whose dimension is equal to the genus of X. Moreover, assuming X has an F-rational point, the elements of Pic 0 (X) are in natural bijection with the F-rational points on J. In particular, with these hypothesis, the class group of K, which is isomorphic to Pic 0 (X), is in bijection with the group of F-rational points on an algebraic variety over a finite field. This gives an alternative more complicated proof of finiteness of the degree 0 class group of a function field. Without the degree 0 condition, the divisor class group won’t be finite. It is an extension of Z by a finite group. 0 → Pic 0 (X) → Pic(X) deg −−→ nZ → 0, where n is the greatest common divisor of the degrees of elements of Pic(X), which is 1 when X has a rational point.
Chapter 22 Exercises ⎛ ⎞ 4 7 2 1. Let A = ⎝2 4 6⎠. 0 0 0 (a) Find invertible integer matrices P and Q such that PAQ is in Smith normal form. (b) What is the group structure of the cokernel of the map Z 3 → Z 3 defined by multiplication by A? 2. Let G be the abelian group generated by x, y, z with relatoins 2x + y = 0 and x − y + 3z = 0. Find a product of cyclic groups that is isomorphic to G. 3. Prove that each of the following rings have infinitely many prime ideals: (a) The integers Z. [Hint: Euclid gave a famous proof of this long ago.] (b) The ring Q[x] of polynomials over Q. (c) The ring Z[x] of polynomials over Z. (d) The ring Z of all algebraic integers. [Hint: Use Zorn’s lemma, which implies that every ideal is contained in a maximal ideal. See, e.g., Prop 1.12 on page 589 of Artin’s Algebra.] 4. (This problem was on the graduate qualifying exam on Tuesday.) Let Z denote the subset of all elements of Q that satisfy a monic polynomial with coefficients in the ring Z of integers. We proved in class that Z is a ring. (a) Show that the ideals (2) and ( √ 2) in Z are distinct. (b) Prove that Z is not Noetherian. 5. Show that neither Z[ √ −6] nor Z[ √ 5] is a unique factorization domain. [Hint: Consider the factorization into irreducible elements of 6 in the first case and 4 in the second. A nonzero element a in a ring R is an irreducible element if it is not a unit and if whenever a = qr, then one of q or r is a unit.] 173
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A Brief Introduction to Classical a
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4 CONTENTS 8 Factoring Primes 49 8.
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6 CONTENTS
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8 CHAPTER 1. PREFACE Acknowledgemen
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10 CHAPTER 2. INTRODUCTION 2.2 What
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12 CHAPTER 2. INTRODUCTION (e) Deep
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Chapter 3 Finitely generated abelia
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1. By permuting rows and columns, m
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Chapter 4 Commutative Algebra We wi
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4.1. NOETHERIAN RINGS AND MODULES 2
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4.1. NOETHERIAN RINGS AND MODULES 2
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Chapter 5 Rings of Algebraic Intege
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5.2. NORMS AND TRACES 27 because Z
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5.2. NORMS AND TRACES 29 elements o
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Chapter 6 Unique Factorization of I
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6.1. DEDEKIND DOMAINS 33 of a gener
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6.1. DEDEKIND DOMAINS 35 Theorem 6.
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Chapter 7 Computing 7.1 Algorithms
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7.2. USING MAGMA 39 > S, P, Q := Sm
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7.2. USING MAGMA 41 r4 + r1 > Minim
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7.2. USING MAGMA 43 [ 1, a, 1/3*(a^
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7.2. USING MAGMA 45 7.2.4 Ideals >
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7.2. USING MAGMA 47 > OK := Maximal
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Chapter 8 Factoring Primes First we
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8.1. FACTORING PRIMES 51 [3, 1, 0,
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8.1. FACTORING PRIMES 53 Theorem 8.
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8.1. FACTORING PRIMES 55 Sketch of
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Chapter 9 Chinese Remainder Theorem
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9.1. THE CHINESE REMAINDER THEOREM
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9.1. THE CHINESE REMAINDER THEOREM
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Chapter 10 Discrimannts, Norms, and
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10.2. DISCRIMINANTS 65 Lemma 10.2.1
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10.4. FINITENESS OF THE CLASS GROUP
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Chapter 11 Computing Class Groups I
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11.1. REMARKS ON COMPUTING THE CLAS
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Chapter 12 Dirichlet’s Unit Theor
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12.1. THE GROUP OF UNITS 79 Lemma 1
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12.2. FINISHING THE PROOF OF DIRICH
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12.2. FINISHING THE PROOF OF DIRICH
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12.3. SOME EXAMPLES OF UNITS IN NUM
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12.4. PREVIEW 89 > time G,phi := Un
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Chapter 13 Decomposition and Inerti
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13.2. DECOMPOSITION OF PRIMES 93 If
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13.2. DECOMPOSITION OF PRIMES 95 Th
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Chapter 14 Decomposition Groups and
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14.1. THE DECOMPOSITION GROUP 99 Th
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14.2. FROBENIUS ELEMENTS 101 Figure
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14.3. GALOIS REPRESENTATIONS AND A
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Part II Adelic Viewpoint 105
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108 CHAPTER 15. VALUATIONS Note tha
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110 CHAPTER 15. VALUATIONS To say t
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112 CHAPTER 15. VALUATIONS Proof. F
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114 CHAPTER 15. VALUATIONS 1 so |u|
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116 CHAPTER 15. VALUATIONS of a val
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118 CHAPTER 16. TOPOLOGY AND COMPLE
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120 CHAPTER 16. TOPOLOGY AND COMPLE
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A Brief Introduction to Classical and Adelic Algebraic ... - William Stein

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