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

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186 INDEX complex n-dimensional representation, 102 conjugation of Frobenius proposition, 100 connected, 124, 124 content, 168 content map is continuous lemma, 168 convex, 68 Corollary A + K and base extension, 159 A + K /K+ has finite measure, 162 OK is Noetherian, 29 compact subset of adeles, 161 complete local field locally compact, 131 completion, norms, and traces, 143 discrimant of number field > 1, 70 extension of complete field is complete, 148 factorization of fractoinal ideals, 35 group as quotient of free groups, 16 norm, trace compatible with towers, 28 norms, traces, and completions, 143 order of inertia group, 99 tensor products and characteristic polynomials, 142 topology on adeles, 157 valuation stays non-archimedean, 120 value set stays same, 120 cyclotomic fields, 88 decomposition group, 97 decomposition groups are conjugate lemma, 97 Dedekind domain, 32 dimension of embedding of field proposition, 63 Dirichlet unit theorem, 77 disconnected, 124 discrete, 109 discrete subgroup of R proposition, 110 discrimant of number field > 1 corollary, 70 discriminant, 65 discriminant of order proposition, 65 divides, 33 elliptic curve, 125 equivalent, 107, 138 equivalent non-archimedean valuations and O’s lemma, 111 equivalent valuations, same topology lemma, 117 essential discriminant divisor, 55 Euler, 122 exactness and Noetherian lemma, 20 extends, 145 extension of complete field is complete corollary, 148 extension of normalized valuation lemma, 151 factorization of pOK lemma, 52 factorization of fractoinal ideals corollary, 35 Faltings theorem, 12 Fermat’s last theorem, 123 field of fractions, 31 finitely generated, 15 finiteness of class group theorem, 67, 171 finiteness of function field class group theorem, 172 fixed field characterization proposition, 98 fractional ideal, 32 fractional ideal is lattice lemma, 69 fractional ideals and formal sums of valuations lemma, 171 fractional ideals theorem, 33 Frobenius element, 100 Galois, 91 Galois conjugates, 27 Gelfand-Tornheim theorem, 113
INDEX 187 global field, 153 group as quotient of free groups corollary, 16 group of units, 77 Haar measure, 132 Haar measure on K ∗ lemma, 135 Haar measure on compact lemma, 132 Hasse, 125 Hasse-Minkowski theorem, 125 Hensel’s lemma, 179 Hilbert Basis theorem, 21 homomorphism, 19 hyperplane embedding lemma, 78 icosahedral, 103 ideals generated by two elements proposition, 60 idele group, 167 ideles are a restricted product lemma, 168 inertia group, 99 inertia group characterization proposition, 100 inertia subgroup, 97 integral ideal, 32 integral ideals of bounded norm lemma, 67 integrally closed in its field of fractions, 31 lattice index, 66 lattices and volumes lemma, 68 Lemma I divides product of primes, 33 I ∩ J = IJ, 57 K + and K ∗ are totally disconnected, 135 Norm(aI), 66 OK span and OK ∩ Q = Z, 27 adic-expansion, 130 any two norms equivalent, 138 base extension of adeles, 159 Blichfeld, 68 characterization of discrete, 111 class group generated by bounded primes, 73 content map is continuous, 168 decomposition groups are conjugate, 97 equivalent non-archimedean valuations and O’s, 111 equivalent valuations, same topology, 117 exactness and Noetherian, 20 extension of normalized valuation, 151 factorization of pOK, 52 fractional ideal is lattice, 69 fractional ideals and formal sums of valuations, 171 Haar measure on K∗ , 135 Haar measure on compact, 132 hyperplane embedding, 78 ideles are a restricted product, 168 integral ideals of bounded norm, 67 lattices and volumes, 68 local compactness of restricted product, 157 matching integers, 155 minimal polynomial of algebraic integer, 25 non-archimedean valuation characterization, 112 open ball is closed, 124 principal ideles are discrete, 168 reduction homomorphism, 129 restricted product, 157 structure of tensor product of fields, 140 subset topology on 1-ideles I1 K , 169 surjection and Noetherian, 21 topological field, 117 trace pairing nondegenerate, 65 valuations on k(t), 116 valuations such that |a| > 1, 153 volume of rings of integers, 69
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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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12.3. 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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130 CHAPTER 17. ADIC NUMBERS: THE F
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Page 136 and 137: 136 CHAPTER 17. ADIC NUMBERS: THE F
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Page 184 and 185: 184 BIBLIOGRAPHY [Fre94] G. Frey (e
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