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

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146CHAPTER 19. EXTENSIONS AND NORMALIZATIONS OF VALUATIONS Existence. We do not give a proof of existence in the general case. Instead we give a proof, which was suggested by Dr. Geyer at the conference out of which [Cas67] arose. It is valid when K is locally compact, which is the only case we will use later. We see at once that the function defined in (19.1.1) satisfies the condition (i) that a ≥ 0 with equality only for a = 0, and (ii) ab = a · b for all a, b ∈ L. The difficult part of the proof is to show that there is a constant C > 0 such that a ≤ 1 =⇒ 1 + a ≤ C. Note that we do not know (and will not show) that · as defined by (19.1.1) is a norm as in Definition 18.1.1, since showing that · is a norm would entail showing that it satisfies the triangle inequality, which is not obvious. Choose a basis b1, . . .,bN for L over K. Let · 0 be the max norm on L, so for a = N i=1 cibi with ci ∈ K we have N a0 = cibi i=1 0 = max{|ci| : i = 1, . . .,N}. (Note: in Cassels’s original article he let · 0 be any norm, but we don’t because the rest of the proof does not work, since we can’t use homogeneity as he claims to do. This is because it need not be possible to find, for any nonzero a ∈ L some element c ∈ K such that ac 0 = 1. This would fail, e.g., if a 0 = |c| for any c ∈ K.) The rest of the argument is very similar to our proof from Lemma 18.1.3 of uniqueness of norms on vector spaces over complete fields. With respect to the · 0 -topology, L has the product topology as a product of copies of K. The function a ↦→ a is a composition of continuous functions on L with respect to this topology (e.g., Norm L/K is the determinant, hence polynomial), hence · defines nonzero continuous function on the compact set S = {a ∈ L : a 0 = 1}. By compactness, there are real numbers δ, ∆ ∈ R>0 such that 0 < δ ≤ a ≤ ∆ for all a ∈ S. For any nonzero a ∈ L there exists c ∈ K such that a0 = |c|; to see this take c to be a ci in the expression a = N i=1 cibi with |ci| ≥ |cj| for any j. Hence a/c0 = 1, so a/c ∈ S and 0 ≤ δ ≤ a/c ≤ ∆. a/c0 Then by homogeneity 0 ≤ δ ≤ a a 0 ≤ ∆.
19.1. EXTENSIONS OF VALUATIONS 147 Suppose now that a ≤ 1. Then a 0 ≤ δ −1 , so as required. 1 + a ≤ ∆ · 1 + a 0 ≤ ∆ · (1 0 + a 0 ) ≤ ∆ · 1 0 + δ −1 = C (say), Example 19.1.3. Consider the extension C of R equipped with the archimedean valuation. The unique extension is the ordinary absolute value on C: x + iy = x 2 + y 2 1/2 . Example 19.1.4. Consider the extension Q2( √ 2) of Q2 equipped with the 2-adic absolute value. Since x 2 − 2 is irreducible over Q2 we can do some computations by working in the subfield Q( √ 2) of Q2( √ 2). > K := NumberField(x^2-2); > K; Number Field with defining polynomial x^2 - 2 over the Rational Field > function norm(x) return Sqrt(2^(-Valuation(Norm(x),2))); end function; > norm(1+a); 1.0000000000000000000000000000 > norm(1+a+1); 0.70710678118654752440084436209 > z := 3+2*a; > norm(z); 1.0000000000000000000000000000 > norm(z+1); 0.353553390593273762200422181049 Remark 19.1.5. Geyer’s existence proof gives (19.1.1). But it is perhaps worth noting that in any case (19.1.1) is a consequence of unique existence, as follows. Suppose L/K is as above. Suppose M is a finite Galois extension of K that contains L. Then by assumption there is a unique extension of | · | to M, which we shall also denote by · . If σ ∈ Gal(M/K), then a σ := σ(a) is also an extension of | · | to M, so · σ = · , i.e., But now σ(a) = a for all a ∈ M. Norm L/K(a) = σ1(a) · σ2(a) · · ·σN(a)
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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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Page 138 and 139: 138 CHAPTER 18. NORMED SPACES AND T
Page 148 and 149: 148CHAPTER 19. EXTENSIONS AND NORMA
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Page 174 and 175: 174 CHAPTER 22. EXERCISES 6. Find t
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Page 184 and 185: 184 BIBLIOGRAPHY [Fre94] G. Frey (e
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