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

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88 CHAPTER 12. DIRICHLET’S UNIT THEOREM [ 1.16761574692758757159598251863681302946987760474899999999995, -0.39284872458139826129179862583435951875841422643044369999996, -0.7747670223461893103041838928024535107114633783181766999998 ] > Logs(u2); [ 0.6435429462288618773851817227686467257757954024463081999999, -1.6402241503223171469101505551700850575583464226669999999999, 0.9966812040934552695249688324014383317825510202205498999998 ] A family of fields with r = 0 (totally complex) is the cyclotomic fields Q(ζn). The degree of Q(ζn) over Q is ϕ(n) and r = 0, so s = ϕ(n)/2 (assuming n > 2). > K := CyclotomicField(11); K; Cyclotomic Field of order 11 and degree 10 > G, phi := UnitGroup(K); > G; Abelian Group isomorphic to Z/22 + Z + Z + Z + Z Defined on 5 generators Relations: 22*G.1 = 0 > u := K!phi(G.2); u; zeta_11^9 + zeta_11^8 + zeta_11^7 + zeta_11^6 + zeta_11^5 + zeta_11^3 + zeta_11^2 + zeta_11 + 1 > Logs(u); [ -1.25656632417872848745322215929976803991663080388899999999969, 0.6517968940331400079717923884685099182823284402303273999999, -0.18533004655986214094922163920197221556431542171819269999999, 0.5202849820300749393306985734118507551388955065272236999998, 0.26981449467537568109995283662137958205972227885009159999993 ] > K!phi(G.3); zeta_11^9 + zeta_11^7 + zeta_11^6 + zeta_11^5 + zeta_11^4 + zeta_11^3 + zeta_11^2 + zeta_11 + 1 > K!phi(G.4); zeta_11^9 + zeta_11^8 + zeta_11^7 + zeta_11^6 + zeta_11^5 + zeta_11^4 + zeta_11^3 + zeta_11^2 + zeta_11 > K!phi(G.5); zeta_11^9 + zeta_11^8 + zeta_11^7 + zeta_11^6 + zeta_11^5 + zeta_11^4 + zeta_11^2 + zeta_11 + 1 How far can we go computing unit groups of cyclotomic fields directly with Magma? > time G,phi := UnitGroup(CyclotomicField(13)); Time: 2.210 > time G,phi := UnitGroup(CyclotomicField(17)); Time: 8.600
12.4. PREVIEW 89 > time G,phi := UnitGroup(CyclotomicField(23)); .... I waited over 10 minutes (usage of 300MB RAM) and gave up. 12.4 Preview In the next chapter we will study extra structure in the case when K is Galois over Q; the results are nicely algebraic, beautiful, and have interesting ramifications. We’ll learn about Frobenius elements, the Artin symbol, decomposition groups, and how the Galois group of K is related to Galois groups of residue class fields. These are the basic structures needed to make any sense of representations of Galois groups, which is at the heart of much of number theory.
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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.
Page 37 and 38: Chapter 7 Computing 7.1 Algorithms
Page 39 and 40: 7.2. USING MAGMA 39 > S, P, Q := Sm
Page 41 and 42: 7.2. USING MAGMA 41 r4 + r1 > Minim
Page 43 and 44: 7.2. USING MAGMA 43 [ 1, a, 1/3*(a^
Page 45 and 46: 7.2. USING MAGMA 45 7.2.4 Ideals >
Page 47 and 48: 7.2. USING MAGMA 47 > OK := Maximal
Page 49 and 50: Chapter 8 Factoring Primes First we
Page 51 and 52: 8.1. FACTORING PRIMES 51 [3, 1, 0,
Page 53 and 54: 8.1. FACTORING PRIMES 53 Theorem 8.
Page 55 and 56: 8.1. FACTORING PRIMES 55 Sketch of
Page 57 and 58: Chapter 9 Chinese Remainder Theorem
Page 59 and 60: 9.1. THE CHINESE REMAINDER THEOREM
Page 61 and 62: 9.1. THE CHINESE REMAINDER THEOREM
Page 63 and 64: Chapter 10 Discrimannts, Norms, and
Page 65 and 66: 10.2. DISCRIMINANTS 65 Lemma 10.2.1
Page 67 and 68: 10.4. FINITENESS OF THE CLASS GROUP
Page 73 and 74: Chapter 11 Computing Class Groups I
Page 75 and 76: 11.1. REMARKS ON COMPUTING THE CLAS
Page 77 and 78: Chapter 12 Dirichlet’s Unit Theor
Page 79 and 80: 12.1. THE GROUP OF UNITS 79 Lemma 1
Page 81 and 82: 12.2. FINISHING THE PROOF OF DIRICH
Page 83 and 84: 12.2. FINISHING THE PROOF OF DIRICH
Page 85 and 86: 12.3. SOME EXAMPLES OF UNITS IN NUM
Page 87: 12.3. SOME EXAMPLES OF UNITS IN NUM
Page 91 and 92: Chapter 13 Decomposition and Inerti
Page 93 and 94: 13.2. DECOMPOSITION OF PRIMES 93 If
Page 95 and 96: 13.2. DECOMPOSITION OF PRIMES 95 Th
Page 97 and 98: Chapter 14 Decomposition Groups and
Page 99 and 100: 14.1. THE DECOMPOSITION GROUP 99 Th
Page 101 and 102: 14.2. FROBENIUS ELEMENTS 101 Figure
Page 103 and 104: 14.3. GALOIS REPRESENTATIONS AND A
Page 105: Part II Adelic Viewpoint 105
Page 108 and 109: 108 CHAPTER 15. VALUATIONS Note tha
Page 110 and 111: 110 CHAPTER 15. VALUATIONS To say t
Page 112 and 113: 112 CHAPTER 15. VALUATIONS Proof. F
Page 114 and 115: 114 CHAPTER 15. VALUATIONS 1 so |u|
Page 116 and 117: 116 CHAPTER 15. VALUATIONS of a val
Page 118 and 119: 118 CHAPTER 16. TOPOLOGY AND COMPLE
Page 130 and 131: 130 CHAPTER 17. ADIC NUMBERS: THE F
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138 CHAPTER 18. NORMED SPACES AND T
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146CHAPTER 19. EXTENSIONS AND NORMA
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154 CHAPTER 20. GLOBAL FIELDS AND A
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168 CHAPTER 21. IDELES AND IDEALS E
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170 CHAPTER 21. IDELES AND IDEALS T
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172 CHAPTER 21. IDELES AND IDEALS P
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174 CHAPTER 22. EXERCISES 6. Find t
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176 CHAPTER 22. EXERCISES 22. (*) S
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178 CHAPTER 22. EXERCISES (a) The p
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180 CHAPTER 22. EXERCISES 60. Find
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182 CHAPTER 22. EXERCISES
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184 BIBLIOGRAPHY [Fre94] G. Frey (e
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186 INDEX complex n-dimensional rep
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188 INDEX lies over, 73 local compa
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190 INDEX valuations such that |a|
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