184 BIBLIOGRAPHY [Fre94] G. Frey (ed.), On Artin’s conjecture for odd 2-dimensional representations, Springer-Verlag, Berlin, 1994, 1585. MR 95i:11001 [Iwa53] K. Iwasawa, On the rings of valuation vec<strong>to</strong>rs, Ann. of Math. (2) 57 (1953), 331–356. MR 14,849a [Lan64] S. Lang, <strong>Algebraic</strong> numbers, Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Al<strong>to</strong>-London, 1964. MR 28 #3974 [Lan80] R. P. Langl<strong>and</strong>s, Base change for GL(2), Prince<strong>to</strong>n University Press, Prince<strong>to</strong>n, N.J., 1980. [Len02] H.W. Lenstra, Jr., Solving the Pell equation, Notices Amer. Math. Soc. 49 (2002), no. 2, 182–192. MR 2002i:11028 [LL93] A.K. Lenstra <strong>and</strong> H.W. Lenstra, Jr. (eds.), The development of the number field sieve, Springer-Verlag, Berlin, 1993. MR 96m:11116 [Mah64] K. Mahler, Inequalities for ideal bases in algebraic number fields, J. Austral. Math. Soc. 4 (1964), 425–448. MR 31 #1243 [SD01] H.P. F. Swinner<strong>to</strong>n-Dyer, A brief guide <strong>to</strong> algebraic number theory, London Mathematical Society Student Texts, vol. 50, Cambridge University Press, Cambridge, 2001. MR 2002a:11117 [Ser73] J-P. Serre, A Course in Arithmetic, Springer-Verlag, New York, 1973, Translated from the French, Graduate Texts in Mathematics, No. 7. [Wei82] A. Weil, Adeles <strong>and</strong> algebraic groups, Progress in Mathematics, vol. 23, Birkhäuser Bos<strong>to</strong>n, Mass., 1982, With appendices by M. Demazure <strong>and</strong> Takashi Ono. MR 83m:10032
Index 1-ideles, 169 I divides product of primes lemma, 33 I ∩ J = IJ lemma, 57 K + <strong>and</strong> K∗ are <strong>to</strong>tally disconnected lemma, 135 N-adic distance, 121 N-adic numbers, 122 N-adic valuation, 121 N-distance is metric proposition, 121 R-module, 19 A + K <strong>and</strong> base extension corollary, 159 A + K /K+ has finite measure corollary, 162 Norm(aI) lemma, 66 OK is Dedekind proposition, 32 OK is Noetherian corollary, 29 OK is a lattice proposition, 28 OK is integrally closed proposition, 31 OK span <strong>and</strong> OK ∩ Q = Z lemma, 27 QN <strong>to</strong>tally disconnected proposition, 124 Z is a PID proposition, 22 Z is a ring proposition, 26 e, f, g proposition, 98 p-adic field, 123 Magma, 9, 10, 26, 31, 35, 37–41, 43, 46, 50, 53, 55, 58, 65, 71, 82, 84, 85, 88, 92, 95, 130, 174, 176, 177 abelian groups structure theorem, 15 adele ring, 158 adic-expansion lemma, 130 algebraic integer, 25 <strong>Algebraic</strong> number theory, 10 185 almost all, 155 any two norms equivalent lemma, 138 archimedean, 110 Artin symbol, 102 ascending chain condition, 20 base extension of adeles lemma, 159 Birch <strong>and</strong> Swinner<strong>to</strong>n-Dyer conjecture, 125 Blichfeld lemma, 68 Blichfeldt’s lemma, 80 Cauchy sequence, 119 characterization of discrete lemma, 111 characterization of integrality proposition, 26 characterization of Noetherian proposition, 20 chinese remainder theorem, 58 class group, 67, 171 class group generated by bounded primes lemma, 73 cokernel, 16 compact quotient of adeles theorem, 160 compact quotient of ideles theorem, 170 compact subset of adeles corollary, 161 compactness of ring of integers theorem, 131 complete, 119, 119 complete embedding theorem, 119 complete local field locally compact corollary, 131 completion, 119 completion, norms, <strong>and</strong> traces corollary, 143
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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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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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120 CHAPTER 16. TOPOLOGY AND COMPLE
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122 CHAPTER 16. TOPOLOGY AND COMPLE
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124 CHAPTER 16. TOPOLOGY AND COMPLE
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126 CHAPTER 16. TOPOLOGY AND COMPLE
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128 CHAPTER 16. TOPOLOGY AND COMPLE
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130 CHAPTER 17. ADIC NUMBERS: THE F
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132 CHAPTER 17. ADIC NUMBERS: THE F
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