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

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96 CHAPTER 13. DECOMPOSITION AND INERTIA GROUPS > R := PolynomialRing(RationalField()); > K := NumberField(x^3-2); > L := NumberField(x^2+3); > M := CompositeFields(K,L)[1]; > O_M := MaximalOrder(M); > a := M!K.1; > b := M!L.1; > O := Order([a,b]); > Index(O_M,O); 24
Chapter 14 Decomposition Groups and Galois Representations 14.1 The Decomposition Group Suppose K is a number field that is Galois over Q with group G = Gal(K/Q). Fix a prime p ⊂ OK lying over p ∈ Z. Definition 14.1.1 (Decomposition group). The decomposition group of p is the subgroup Dp = {σ ∈ G : σ(p) = p} ≤ G. (Note: The decomposition group is called the “splitting group” in Swinnerton- Dyer. Everybody I know calls it the decomposition group, so we will too.) Let Fp = OK/p denote the residue class field of p. In this section we will prove that there is a natural exact sequence 1 → Ip → Dp → Gal(Fp/Fp) → 1, where Ip is the inertia subgroup of Dp, and #Ip = e. The most interesting part of the proof is showing that the natural map Dp → Gal(Fp/Fp) is surjective. We will also discuss the structure of Dp and introduce Frobenius elements, which play a crucial roll in understanding Galois representations. Recall that G acts on the set of primes p lying over p. Thus the decomposition group is the stabilizer in G of p. The orbit-stabilizer theorem implies that [G : Dp] equals the orbit of p, which by Theorem 13.2.2 equals the number g of primes lying over p, so [G : Dp] = g. Lemma 14.1.2. The decomposition subgroups Dp corresponding to primes p lying over a given p are all conjugate in G. Proof. We have τ(σ(τ −1 (p))) = p if and only if σ(τ −1 (p)) = τ −1 p. Thus τστ −1 ∈ Dp if and only if σ ∈ D τ −1 p, so τ −1 Dpτ = D τ −1 p. The lemma now follows because, by Theorem 13.2.2, G acts transitively on the set of p lying over p. 97
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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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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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