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

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56 CHAPTER 8. FACTORING PRIMES > K := NumberField(x^3 + x^2 - 2*x + 8); > OK := MaximalOrder(K); > Factorization(2*OK); [ , , ] Thus 2OK = p1p2p3, with the pi distinct. Moreover, one can check that OK/pi ∼ = F2. If OK = Z[a] for some a ∈ OK with minimal polynomial g, then g(x) ∈ F2[x] must be a product of three distinct linear factors, which is impossible.
Chapter 9 Chinese Remainder Theorem In this section we will prove the Chinese Remainder Theorem for rings of integers, deduce several surprising and useful consequences, then learn about discriminants, and finally norms of ideals. We will also define the class group of OK and state the main theorem about it. The tools we develop here illustrate the power of what we have already proved about rings of integers, and will be used over and over again to prove other deeper results in algebraic number theory. It is essentially to understand everything we discuss in this chapter very well. 9.1 The Chinese Remainder Theorem Recall that the Chinese Remainder Theorem from elementary number theory asserts that if n1, . . .,nr are integers that are coprime in pairs, and a1, . . .,ar are integers, then there exists an integer a such that a ≡ ai (mod ni) for each i = 1, . . .,r. In terms of rings, the Chinese Remainder Theorem asserts that the natural map Z/(n1 · · ·nr)Z → (Z/n1Z) ⊕ · · · ⊕ (Z/nrZ) is an isomorphism. This result generalizes to rings of integers of number fields. Lemma 9.1.1. If I and J are coprime ideals in OK, then I ∩ J = IJ. Proof. The ideal I ∩ J is the largest ideal of OK that is divisible by (contained in) both I and J. Since I and J are coprime, I ∩ J is divisible by IJ, i.e., I ∩ J ⊂ IJ. By definition of ideal IJ ⊂ I ∩ J, which completes the proof. Remark 9.1.2. This lemma is true for any ring R and ideals I, J ⊂ R such that I + J = R. For the general proof, choose x ∈ I and y ∈ J such that x + y = 1. If c ∈ I ∩ J then c = c · 1 = c · (x + y) = cx + cy ∈ IJ + IJ = IJ, so I ∩ J ⊂ IJ, and the other inclusion is obvious by definition. 57
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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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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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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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