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

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176 CHAPTER 22. EXERCISES 22. (*) Suppose O is an order in the ring of integers OK of a number field. Is every ideal in O necessarily generated by two elements? (Answer: No. Challenge: Given an example.) 23. Find representative ideals for each element of the class group of Q( √ −23). Illustrate how to use the Minkowski bound to prove that your list of representatives is complete. 24. Suppose O is an order in the ring of integers OK of a number field. Is every ideal in O necessarily generated by two elements? 25. Let K be a number field of degree n > 1 with s pairs of complex conjugate embeddings. Prove that π s · 4 nn > 1. n! 26. Do the exercise on page 19 of Swinnerton-Dyer, which shows that the quantity Cr,s in the finiteness of class group theorem can be taken to be 4 s n! π nn. 27. Let α denote a root of x3 − x + 2 and let K = Q(α). Show that OK = Z[α] and that K has class number 1 (don’t just read this off from the output of the Magma commands MaximalOrder and ClassNumber). [Hint: consider the square factors of the discriminant of x3 −x+2 and show that 1 2 (a+bα+cα2 ) is an algebra integer if and only if a, b, and c are all even.] 28. If S is a closed, bounded, convex, symmetric set in R n with Vol(S) ≥ m2 n , for some positive integer m, show that S contains at least 2m nonzero points in Z n . 29. Prove that any finite subgroup of the multiplicative group of a field is cyclic. 30. For a given number field K, which seems more difficult for Magma to compute, the class groups or explicit generators for the group of units? It is very difficult (but not impossible) to not get full credit on this problem. Play around with some examples, see what seems more difficult, and justify your response with examples. (This problem might be annoying to do using the Magma web page, since it kills your Magma job after 30 seconds. Feel free to request a binary of Magma from me, or an account on MECCAH (Mathematics Extreme Computation Cluster at Harvard).) 31. (a) Prove that there is no number field K such that UK ∼ = Z/10Z. (b) Is there a number field K such that UK ∼ = Z × Z/6Z? 32. Prove that the rank of UK is unbounded as K varies over all number fields. 33. Let K = Q(ζ5). (a) Show that r = 0 and s = 2.
177 (b) Find explicitly generators for the group of units of UK (you can use Magma for this). (c) Draw an illustration of the log map ϕ : UK → R 2 , including the hyperplane x1+x2 = 0 and the lattice in the hyperplane spanned by the image of UK. 34. Find the group of units of Q(ζn) as an abstract group as a function of n. (I.e., find the number of cyclic factors and the size of the torsion subgroup. You do not have to find explicit generators!) 35. Let K = Q(a), where a is a root x 3 − 3x + 1. (a) Show that r = 3. (b) Find explicitly the log embedding of UK into a 2-dimensional hyperplane in R 3 , and draw a picture. 36. Prove that if K is a quadratic field and the torsion subgroup of UK has order bigger than 2, then K = Q( √ −3) or K = Q( √ −1). 37. A Salem number is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. By any method (including “google”), give two examples of Salem numbers. 38. Let p ∈ Z and let K be a number field. Show that Norm K/Q(pOK) = p [K:Q] . 39. A totally real number field is a number field in which all embeddings into C have image in R. Prove there are totally real number fields of degree p, for every prime p. [Hint: Let ζn denote a primitive nth root of unity. For n ≥ 3, show that Q(ζn + 1/ζn) is totally real of degree ϕ(n)/2. Now prove that ϕ(n)/2 can be made divisible by any prime.] 40. Give an example of a number field K/Q and a prime p such that the ei in the factorization of pOK are not all the same. 41. Let K be a number field. Give the “simplest” proof you can think of that there are only finitely many primes that ramify (i.e., have some ei > 1) in K. [The meaning of “simplest” is a matter of taste.] 42. Give examples to show that for K/Q a Galois extension, the quantity e can be arbirarily large and f can be arbitrarily large. 43. Suppose K/Q is Galois and p is a prime such that pOK is also prime (i.e., p is inert in K). Show that Gal(K/Q) is a cyclic group. 44. (Problem 7, page 116, from Marcus Number Fields) For each of the following, find a prime p and quadratic extensions K and L of Q that illustrates the assertion:
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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 126 and 127: 126 CHAPTER 16. TOPOLOGY AND COMPLE
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Page 184 and 185: 184 BIBLIOGRAPHY [Fre94] G. Frey (e
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