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

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178 CHAPTER 22. EXERCISES (a) The prime p can be totally ramified in K and L without being totally ramified in KL. (b) The fields K and L can each contain unique primes lying over p while KL does not. (c) The prime p can be inert in K and L without being inert in KL. (d) The residue field extensions of Fp can be trivial for K and L without being trivial for KL. 45. Let S3 by the symmetric group on three symbols, which has order 6. (a) Observe that S3 ∼ = D3, where D3 is the dihedral group of order 6, which is the group of symmetries of an equilateral triangle. (b) Use (45a) to write down an explicit embedding S3 ↩→ GL2(C). (c) Let K be the number field Q( 3√ 2, ω), where ω 3 = 1 is a nontrivial cube root of unity. Show that K is a Galois extension with Galois group isomorphic to S3. (d) We thus obtain a 2-dimensional irreducible complex Galois representation ρ : Gal(Q/Q) → Gal(K/Q) ∼ = S3 ⊂ GL2(C). Compute a representative matrix of Frobp and the characteristic polynomial of Frobp for p = 5, 7, 11, 13. 46. Let K = Q( √ 2, √ 3, √ 5, √ 7). Show that K is Galois over Q, compute the Galois group of K, and compute Frob37. 47. Let k be any field. Prove that the only nontrivial valuations on k(t) which are trivial on k are equivalent to the valuation (15.3.3) or (15.3.4) of page 115. 48. A field with the topology induced by a valuation is a topological field, i.e., the operations sum, product, and reciprocal are continuous. 49. Give an example of a non-archimedean valuation on a field that is not discrete. 50. Prove that the field Qp of p-adic numbers is uncountable. 51. Prove that the polynomial f(x) = x 3 − 3x 2 + 2x + 5 has all its roots in Q5, and find the 5-adic valuations of each of these roots. (You might need to use Hensel’s lemma, which we don’t discuss in detail in this book. See [Cas67, App. C].) 52. In this problem you will compute an example of weak approximation, like I did in the Example 16.3.3. Let K = Q, let | · | 7 be the 7-adic absolute value, let | · | 11 be the 11-adic absolute value, and let | · | ∞ be the usual archimedean absolute value. Find an element b ∈ Q such that |b − ai| i < 1 10 , where a7 = 1, a11 = 2, and a∞ = −2004.
179 53. Prove that −9 has a cube root in Q10 using the following strategy (this is a special case of Hensel’s Lemma, which you can read about in an appendix to Cassel’s article). (a) Show that there is an element α ∈ Z such that α 3 ≡ 9 (mod 10 3 ). (b) Suppose n ≥ 3. Use induction to show that if α1 ∈ Z and α 3 ≡ 9 (mod 10 n ), then there exists α2 ∈ Z such that α 3 2 ≡ 9 (mod 10n+1 ). (Hint: Show that there is an integer b such that (α1 + b · 10 n ) 3 ≡ 9 (mod 10 n+1 ).) (c) Conclude that 9 has a cube root in Q10. 54. Compute the first 5 digits of the 10-adic expansions of the following rational numbers: 13 2 , 1 389 , 17 , 19 the 4 square roots of 41. 55. Let N > 1 be an integer. Prove that the series ∞ (−1) n+1 n! = 1! − 2! + 3! − 4! + 5! − 6! + · · · . n=1 converges in QN. 56. Prove that −9 has a cube root in Q10 using the following strategy (this is a special case of “Hensel’s Lemma”). (a) Show that there is α ∈ Z such that α 3 ≡ 9 (mod 10 3 ). (b) Suppose n ≥ 3. Use induction to show that if α1 ∈ Z and α 3 ≡ 9 (mod 10 n ), then there exists α2 ∈ Z such that α 3 2 ≡ 9 (mod 10n+1 ). (Hint: Show that there is an integer b such that (α1 + b10 n ) 3 ≡ 9 (mod 10 n+1 ).) (c) Conclude that 9 has a cube root in Q10. 57. Let N > 1 be an integer. (a) Prove that QN is equipped with a natural ring structure. (b) If N is prime, prove that QN is a field. 58. (a) Let p and q be distinct primes. Prove that Qpq ∼ = Qp × Qq. (b) Is Q p 2 isomorphic to either of Qp × Qp or Qp? 59. Prove that every finite extension of Qp “comes from” an extension of Q, in the following sense. Given an irreducible polynomial f ∈ Qp[x] there exists an irreducible polynomial g ∈ Q[x] such that the fields Qp[x]/(f) and Qp[x]/(g) are isomorphic. [Hint: Choose each coefficient of g to be sufficiently close to the corresponding coefficient of f, then use Hensel’s lemma to show that g has a root in Qp[x]/(f).]
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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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Page 128 and 129: 128 CHAPTER 16. TOPOLOGY AND COMPLE
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Page 184 and 185: 184 BIBLIOGRAPHY [Fre94] G. Frey (e
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A Brief Introduction to Classical and Adelic Algebraic ... - William Stein

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