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

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10 CHAPTER 2. INTRODUCTION 2.2 What is algebraic number theory? A number field K is a finite algebraic extension of the rational numbers Q. Every such extension can be represented as all polynomials in an algebraic number α: m K = Q(α) = anα n : an ∈ Q . n=0 Here α is a root of a polynomial with coefficients in Q. Algebraic number theory involves using techniques from (mostly commutative) algebra and finite group theory to gain a deeper understanding of number fields. The main objects that we study in algebraic number theory are number fields, rings of integers of number fields, unit groups, ideal class groups,norms, traces, discriminants, prime ideals, Hilbert and other class fields and associated reciprocity laws, zeta and L-functions, and algorithms for computing each of the above. 2.2.1 Topics in this book These are some of the main topics that are discussed in this book: • Rings of integers of number fields • Unique factorization of ideals in Dedekind domains • Structure of the group of units of the ring of integers • Finiteness of the group of equivalence classes of ideals of the ring of integers (the “class group”) • Decomposition and inertia groups, Frobenius elements • Ramification • Discriminant and different • Quadratic and biquadratic fields • Cyclotomic fields (and applications) • How to use a computer to compute with many of the above objects (both algorithms and actual use of PARI and Magma). • Valuations on fields • Completions (p-adic fields) • Adeles and Ideles
2.3. SOME APPLICATIONS OF ALGEBRAIC NUMBER THEORY 11 Note that we will not do anything nontrivial with zeta functions or L-functions. This is to keep the prerequisites to algebra, and so we will have more time to discuss algorithmic questions. Depending on time and your inclination, I may also talk about integer factorization, primality testing, or complex multiplication elliptic curves (which are closely related to quadratic imaginary fields). 2.3 Some applications of algebraic number theory The following examples are meant to convince you that learning algebraic number theory now will be an excellent investment of your time. If an example below seems vague to you, it is safe to ignore it. 1. Integer factorization using the number field sieve. The number field sieve is the asymptotically fastest known algorithm for factoring general large integers (that don’t have too special of a form). Recently, in December 2003, the number field sieve was used to factor the RSA-576 $10000 challenge: 1881988129206079638386972394616504398071635633794173827007 . . . . . .6335642298885971523466548531906060650474304531738801130339 . . . . . .6716199692321205734031879550656996221305168759307650257059 = 39807508642406493739712550055038649119906436234252670840 . . . . . .6385189575946388957261768583317 ×47277214610743530253622307197304822463291469530209711 . . . . . .6459852171130520711256363590397527 (The . . . indicates that the newline should be removed, not that there are missing digits.) For more information on the NFS, see the paper by Lenstra et al. on the Math 129 web page. 2. Primality test: Agrawal and his students Saxena and Kayal from India recently (2002) found the first ever deterministic polynomial-time (in the number of digits) primality test. There methods involve arithmetic in quotients of (Z/nZ)[x], which are best understood in the context of algebraic number theory. For example, Lenstra, Bernstein, and others have done that and improved the algorithm significantly. 3. Deeper point of view on questions in number theory: (a) Pell’s Equation (x 2 −dy 2 = 1) =⇒ Units in real quadratic fields =⇒ Unit groups in number fields (b) Diophantine Equations =⇒ For which n does x n + y n = z n have a nontrivial solution in Q( √ 2)? (c) Integer Factorization =⇒ Factorization of ideals (d) Riemann Hypothesis =⇒ Generalized Riemann Hypothesis
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
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Page 15 and 16: Chapter 3 Finitely generated abelia
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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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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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