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

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182 CHAPTER 22. EXERCISES
Bibliography [ABC + ] B. Allombert, K. Belabas, H. Cohen, X. Roblot, and I. Zakharevitch, PARI/GP, http://pari.math.u-bordeaux.fr/. [Art59] E. Artin, Theory of algebraic numbers, Notes by Gerhard Würges from lectures held at the Mathematisches Institut, Göttingen, Germany, in the Winter Semester, vol. 1956/7, George Striker, Schildweg 12, Göttingen, 1959. MR 24 #A1884 [Art91] M. Artin, Algebra, Prentice Hall Inc., Englewood Cliffs, NJ, 1991. MR 92g:00001 [AT90] E. Artin and J. Tate, Class field theory, second ed., Advanced Book Classics, Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1990. MR 91b:11129 [BCP97] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265, Computational algebra and number theory (London, 1993). MR 1 484 478 [Cas67] J. W.S. Cassels, Global fields, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 42– 84. [Cas91] , Lectures on elliptic curves, London Mathematical Society Student Texts, vol. 24, Cambridge University Press, Cambridge, 1991. MR 92k:11058 [Coh93] H. Cohen, A course in computational algebraic number theory, Springer- Verlag, Berlin, 1993. MR 94i:11105 [Cp86] J. W.S. Cassels and A. Fröhlich (eds.), Algebraic number theory, London, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], 1986, Reprint of the 1967 original. [EH00] D. Eisenbud and J. Harris, The geometry of schemes, Springer-Verlag, New York, 2000. MR 2001d:14002 183
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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
Page 172 and 173: 172 CHAPTER 21. IDELES AND IDEALS P
Page 174 and 175: 174 CHAPTER 22. EXERCISES 6. Find t
Page 176 and 177: 176 CHAPTER 22. EXERCISES 22. (*) S
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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 188 and 189: 188 INDEX lies over, 73 local compa
Page 190: 190 INDEX valuations such that |a|
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