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

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36 CHAPTER 6. UNIQUE FACTORIZATION OF IDEALS > [K!b : b in Basis(OK)]; [ 1, K.1 // this is sqrt(-6) ] > Factorization(6*OK); [ , ] The output means that (6) = (2, 2 + √ −6) 2 · (3, 3 + √ −6) 2 , where each of the ideals (2, 2 + √ −6) and (3, 3 + √ −6) is prime. I will discuss algorithms for computing such a decomposition in detail, probably next week. The first idea is to write (6) = (2)(3), and hence reduce to the case of writing the (p), for p ∈ Z prime, as a product of primes. Next one decomposes the Artinian ring OK ⊗ Fp as a product of local Artinian rings.
Chapter 7 Computing 7.1 Algorithms for Algebraic Number Theory I think the best overall reference for algorithms for doing basic algebraic number theory computations is [Coh93]. Our main long-term algorithmic goals for this book (which we won’t succeed at reaching) are to understand good algorithms for solving the following problems in particular cases: • Ring of integers: Given a number field K (by giving a polynomial), compute the full ring OK of integers. • Decomposition of primes: Given a prime number p ∈ Z, find the decomposition of the ideal pOK as a product of prime ideals of OK. • Class group: Compute the group of equivalence classes of nonzero ideals of OK, where I and J are equivalent if there exists α ∈ OK such that IJ −1 = (α). • Units: Compute generators for the group of units of OK. As we will see, somewhat surprisingly it turns out that algorithmically by far the most time-consuming step in computing the ring of integers OK seems to be to factor the discriminant of a polynomial whose root generates the field K. The algorithm(s) for computing OK are quite complicated to describe, but the first step is to factor this discriminant, and it takes much longer in practice than all the other complicated steps. 7.2 Using Magma This section is a first introduction to Magma for algebraic number theory. Magma is a good general purpose package for doing algebraic number theory computations. You can use it via the web pagehttp://modular.fas.harvard.edu/calc. Magma is not free, but student discounts are available. 37
Page 1: A Brief Introduction to Classical a
Page 4 and 5: 4 CONTENTS 8 Factoring Primes 49 8.
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Page 8 and 9: 8 CHAPTER 1. PREFACE Acknowledgemen
Page 10 and 11: 10 CHAPTER 2. INTRODUCTION 2.2 What
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Page 15 and 16: Chapter 3 Finitely generated abelia
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Page 21 and 22: 4.1. NOETHERIAN RINGS AND MODULES 2
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Page 27 and 28: 5.2. NORMS AND TRACES 27 because Z
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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
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Page 47 and 48: 7.2. USING MAGMA 47 > OK := Maximal
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Page 57 and 58: Chapter 9 Chinese Remainder Theorem
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Page 79 and 80: 12.1. THE GROUP OF UNITS 79 Lemma 1
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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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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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A Brief Introduction to Classical and Adelic Algebraic ... - William Stein

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