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

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188 INDEX lies over, 73 local compactness of restricted product lemma, 157 local-to-global principal, 125 locally compact, 131 matching integers lemma, 155 metric, 119 minimal polynomial, 25 minimal polynomial of algebraic integer lemma, 25 Minkowski, 125 multiplicativity of ideal norm proposition, 66 N-adic numbers, 121 totally disconnected, 124 nilpotent, 180 Noetherian, 19 Noetherian equals finitely generated proposition, 21 non-archimedean, 110 non-archimedean valuation characterization lemma, 112 nontrivial solution, 125 norm, 27, 66, 137 norm and trace proposition, 28 norm, trace compatible with towers corollary, 28 normalized, 133 norms, traces, and completions corollary, 143 number field, 10, 26 open ball is closed lemma, 124 open balls, 117 open problem solvability of plane cubics, 125 order, 26, 110 order of inertia group corollary, 99 Ostrowski theorem, 113 prime ideal factorization theorem, 53 principal adeles, 158 principal ideles, 168 principal ideles are discrete lemma, 168 product formula theorem, 154 product measure, 158 product of extensions theorem, 152 properties of Haar measure theorem, 133 Proposition N-distance is metric, 121 OK is Dedekind, 32 OK is a lattice, 28 OK is integrally closed, 31 QN totally disconnected, 124 Z is a PID, 22 Z is a ring, 26 e, f, g, 98 characterization of integrality, 26 characterization of Noetherian, 20 conjugation of Frobenius, 100 dimension of embedding of field, 63 discrete subgroup of R, 110 discriminant of order, 65 fixed field characterization, 98 ideals generated by two elements, 60 inertia group characterization, 100 multiplicativity of ideal norm, 66 Noetherian equals finitely generated, 21 norm and trace, 28 same topology implies equivalent valuations, 118 Smith normal form, 16 structure of p n /p n+1 , 60 subgroup of free group, 15 triangle inequality, 108 unit norm characterization, 78 radical, 55 reduction homomorphism lemma, 129 reduction of Galois group theorem, 99 residue class degree, 92 restricted product lemma, 157
INDEX 189 restricted topological product, 157 ring of N-adic numbers, 122 ring of integers, 26, 111 Salem number, 177 same topology implies equivalent valuations proposition, 118 Selmer, 125 Selmer curve, 125 separable, 140 Shafarevich-Tate group, 125 short exact sequence, 19 Smith normal form, 15 Smith normal form proposition, 16 strong approximation theorem, 165 structure of p n /p n+1 proposition, 60 structure of abelian groups theorem, 15 structure of tensor product of fields lemma, 140 structure theorem, 15 subgroup of free group proposition, 15 submodule, 19 subset topology on 1-ideles I 1 K lemma, 169 surjection and Noetherian lemma, 21 symmetric about the origin, 68 tensor product topology, 140 tensor products and characteristic polynomials corollary, 142 Theorem chinese remainder, 58 compact quotient of adeles, 160 compact quotient of ideles, 170 compactness of ring of integers, 131 complete embedding, 119 Dirichlet unit, 77 Faltings, 12 finiteness of class group, 67, 171 finiteness of function field class group, 172 fractional ideals, 33 Gelfand-Tornheim, 113 Hasse-Minkowski, 125 Hilbert Basis, 21 Ostrowski, 113 prime ideal factorization, 53 product formula, 154 product of extensions, 152 properties of Haar measure, 133 reduction of Galois group, 99 strong approximation, 165 structure of abelian groups, 15 transitive Galois action, 93 unique ideal factorization, 35 uniqueness of valuation extension, 145 valuation extensions, 149 valuations on Q, 113 weak approximation, 125 topological field, 117 topological field lemma, 117 topology on adeles corollary, 157 totally disconnected, 124 trace, 27 trace pairing, 64 trace pairing nondegenerate lemma, 65 transitive Galois action theorem, 93 triangle inequality proposition, 108 trivial valuation, 107 unique ideal factorization theorem, 35 uniqueness of valuation extension theorem, 145 unit norm characterization proposition, 78 valuation, 107, 129 discrete, 109 equivalence of, 107 valuation extensions theorem, 149 valuation stays non-archimedean corollary, 120 valuations on Q theorem, 113 valuations on k(t) lemma, 116
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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 138 and 139: 138 CHAPTER 18. NORMED SPACES AND T
Page 146 and 147: 146CHAPTER 19. EXTENSIONS AND NORMA
Page 154 and 155: 154 CHAPTER 20. GLOBAL FIELDS AND A
Page 168 and 169: 168 CHAPTER 21. IDELES AND IDEALS E
Page 170 and 171: 170 CHAPTER 21. IDELES AND IDEALS T
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Page 174 and 175: 174 CHAPTER 22. EXERCISES 6. Find t
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Page 184 and 185: 184 BIBLIOGRAPHY [Fre94] G. Frey (e
Page 186 and 187: 186 INDEX complex n-dimensional rep
Page 190: 190 INDEX valuations such that |a|
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