Algebraic Number Theory

by Edwin Weiss

Overview

Careful organization and clear, detailed proofs characterize this methodical, self-contained exposition of basic results of classical algebraic number theory from a relatively modem point of view. This volume presents most of the number-theoretic prerequisites for a study of either class field theory (as formulated by Artin and Tate) or the contemporary treatment of analytical questions (as found, for example, in Tate's thesis).
Although concerned exclusively with algebraic number fields, this treatment features axiomatic formulations with a considerable range of applications. Modem abstract techniques constitute the primary focus. Topics include introductory materials on elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields.
Subjects correspond to those usually covered in a one-semester, graduate level course in algebraic number theory, making this book ideal either for classroom use or as a stimulating series of exercises for mathematically minded individuals.

Product Details

ISBN-13:	9780486154367
Publisher:	Dover Publications
Publication date:	01/27/2012
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	288
File size:	33 MB
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Algebraic Number Theory

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By Edwin Weiss

Dover Publications, Inc.

Copyright © 2014 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-15436-7

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Contents

Preface,
References,
Chapter 1 Elementary Valuation Theory,
Chapter 2 Extension of Valuations,
Chapter 3 Local Fields,
Chapter 4 Ordinary Arithmetic Fields,
Chapter 5 Global Fields,
Chapter 6 Quadratic Fields,
Chapter 7 Cyclotomic Fields,
Symbols and Notation,
Index,


CHAPTER 1

Elementary Valuation Theory

The central feature of the subject commonly known as algebraic number theory is the problem of factorization in an algebraic number field, where by an algebraic number field we mean a finite extension of the rational field Q. Of course, it will take some time before the full meaning of this statement will become apparent.

There are several distinct approaches to our subject matter, and we have chosen to emphasize the valuation-theoretic approach. In this chapter, we begin to learn the language.

1-1. Valuations and Prime Divisors

Let us begin with a definition. A valuation of the field F is a function φ from F into the nonnegative reals such that

(i) φ(a) = 0 [??] a = 0

(ii) φ(ab) = φ(a)φ(b)

(iii) There exists a real constant C such that

φ(a) ≤ 1 [??] φ(1 + α) ≤ C

Some obvious examples of valuations are the following: (1) Let F denote the field of real numbers R or the field of complex numbers C, and put φ(a) = |a|, C = 2. It may be remarked that the original impetus for the study of valuations of an arbitrary field arose from this example —that is, as a generalization of the notion of absolute value. (2) Let F be any field, and put φ(0) = 0, φ(a) = 1 for a ≠ 0 ε F. This is known as the trivial valuation of F.

In view of the numerous examples that will be considered later, let us be content for the moment with the simple examples above. The impatient reader may turn to Section 1-4.

A valuation determines a homomorphism of F*, the multiplicative group of F, into the multiplicative group of positive real numbers (φ:F* ->R>0); therefore, we have φ(1) = 1, φ(a-1) = 1/φ(a), and φ(b/a) = φ(b)/(φ(a) for a ≠ 0. Also, since φ(1) = φ(-1)φ(-1), it follows that φ(-l) = 1 and φ(-a) = φ(1)φ(a) = φ(a). It is also immediate that if ζ is any root of unity belonging to F then φ(ζ) = 1 and that a finite field admits only the trivial valuation.

1-1-1. Proposition. In the definition of a valuation, (iii) may be replaced by

(iii') φ(a + b) ≤ C max {φ(a), φ(b)}

Proof. (iii) [??] (iii'). Suppose φ(a) = max {φ(a), φ(b)}. If φ(a) = 0 then a = b = 0 and (iii') holds. If φ(a) ≠ 0, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iii') [??] (iii). If φ(a) ≤ 1, then

φ(1 + a) ≤ C max {φ(1), φ(a)} = C

In the more usual definition of valuation, condition (iii) is replaced by the so-called triangle inequality

φ(a + b) ≤ φ(a) + φ(b)

We shall soon see that, for all practical purposes, our valuations may be taken to satisfy the triangle inequality.

1-1-2. Proposition. A valuation φ of F determines a Hausdorff topology Tφ on F. For each a ε F, a fundamental system of neighborhoods of a is given by the set of all

U(a, ε) = {b ε F|φ(a - b) < ε}

Proof. Let Δ denote the diagonal of F × F, and for subsets S and T of F × F let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For each ε > 0 we put

U(ε) = {(a, b) ε F X F |φ(a - b) < φ}

From the properties of φ, we have

(1) U(ε) [contains] Δ

(2) U(ε1) [intersection] U(ε2) = U(ε) where ε = min (ε1, ε2)

(3) U(ε-1 = U(ε)

(4) U(ε/C) º U(ε/C) [subset] U(ε)

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In fact, (1), (2), and (3) are trivial; (5) is clear since φ(a - b) = 0 [??] a = b; finally, the assertion that (a, b) belongs to the left side of (4) means that there exists c ε F with φ(a - c) < ε/C and φ(c - b) < ε/C, so that φ(a - b) < ε and (4) holds.

Properties 1 to 4 guarantee that {U(ε)} is a filter base on F × F. This filter base defines a uniform structure on F, and according to (5) the uniform structure is separated. Therefore, the topology of F "deduced" from the uniform structure in the canonical way has fundamental neighborhoods as described and is a Hausdorff topology. There is no particular significance attached to the method by which we have arrived at the topology of F. It is based on facts and notation as found in N. Bourbaki's "Topologie Générale" (Hermann & Cie, Paris, 1940).

1-1-3. Corollary. Let φ be a valuation of F, and let {xn} be a sequence in F; then

xn -> 0 for Tφ [??] φ(xn) -> 0

We say that two valuations φ1 and φ2 are equivalent (and denote this by φ1 ~ φ2) when they determine the same topology on F—that is, if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The equivalence classes with respect to this equivalence relation are called prime divisors of F and are denoted by P, Q, etc. The prime divisor to which the trivial valuation of F belongs is called the trivial prime divisor; all the others are nontrivial prime divisors.

Let φ be a valuation of F with constant C, and let α > 0 be a real number; then the function φα on F given by φα(a) = [φ(a)]α is a valuation of F with constant Cα.

1-1-4. Theorem. Let φ1, φ2 be nontrivial valuations of F, and let a denote an arbitrary element of F; then the following statements are

(1) φ2 = φa1 with α > 0

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(4) φ1(a) < 1 [??] φ2(a) < 1

(5) φ1(a) ≤ 1 [??] φ2(a) ≤ 1

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. (1) [??] (2). For i = 1, 2 we write

Ui(ε) = {(a, b) ε F X F |φi(a - b) < ε}

Then U2(εa) = U1(ε), so that

{U2(ε)|ε > 0} = {U1(ε)|ε > 0}

Thus the uniform structures are equivalent, and the topologies are the same—that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(2) [??] (3). This is trivial since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] stronger than [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] means that every open set for the topology [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is also open for the topology [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(3) [??] (4). Making use of (1-1-3), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(4) [??] (5). Since φ1 is nontrivial, there exists b ε F* such that φ1(b) ≠ 1, and we may then take φ1(b) < 1. Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the same way, φ2(a-l) ≤ 1, and we conclude that φ2(a) = 1.

(5) [??] (6). Since φ2 is nontrivial, there exists c ε F* such that 0 < φ2(c) < 1, and then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus (4) holds, and as in the proof that (4) [??] (5) we have

φ1(a) = 1 [??] φ2(a) = 1

Of course, φ1(a) > 1 [??] φ1(a-1) < 1 [??] φ2(a-1) < 1 [??] φ2(a) >1. The validity of (6) is now clear.

(6) [??] (1). Fix a ε F such that φ1(a) > 1. Then φ2(a) > 1, and we may put

α = log φ2(a)/log φ1(a) > 0

We show that φ2 = φa1; for this it suffices to show that if for each b ε F* we write γi = [log φi(b)]/log φi(a)] (i = 1, 2) then γ1 = γ2. Let r = m/n with n > 0 denote a rational number; then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1-1-5. Corollary. Let P be a prime divisor of F; then for any φ ε P

P = {φa|α > 0}

Proof. (1-1-4) takes care of the case where P is nontrivial. Since it is easy to see that the valuation φ of F is trivial if and only if Tφ is the discrete topology, it follows that a trivial prime divisor P consists solely of the trivial valuation; thus, the assertion holds in this case also.

In order to clarify the connection between our valuations and those satisfying the triangle inequality, it is convenient to introduce a simple definition. Given a valuation φ, we define ||φ|| (the norm of φ) by ||φ|| = inf C, where C runs over all constants that may be used in (iii) of the definition of valuation. ||φ|| is the smallest possible value for C, and (1-1-1) applies with C = ||φ||. For any real number α > 0, it follows immediately that ||φα|| = ||α||α. In view of (1-1-5) we see that, given a prime divisor P of F, there exists φ ε P with ||φ|| ≤ 2. We may also note that, for any φ, ||φ|| ≥ 1; in fact, 1 = φ(1) = φ(l + 0) < ||φ||.

1-1-6. Exercise. Suppose that φ is a valuation for which condition (iii) is replaced by the triangle inequality. Give an example to show that under such circumstances φα need not be a valuation (in the sense that it need not satisfy the triangle inequality).

1-1-7. Lemma. Let φ be a valuation of F; then

||φ|| ≤ 2 [??] φ(a1 + ··· + an) ≤ 2n max {phi]([a1), ..., φ(an)}

Proof. By (4-9-1) we have φ(a1 + a2) ≤ 2 max {φ(a1), φ(a2)}, and by induction

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(Continues...)

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Table of Contents

Preface; References
Chapter 1. Elementary Valuation Theory
1.1 Valuations and Prime Divisors
1.2 The Approximation Theorem
1.3 Archimedean and Nonarchimedean Prime Divisors
1.4 The Prime Divisors of Q
1.5 Fields with a Discrete Prime Divisor
1.6 e and f
1.7 Completions
1.8 The Theorem of Ostrowski
1.9 Complete Fields with Discrete Prime Divisor; Exercises
Chapter 2. Extension of Valuations
2.1 Uniqueness of Extensions (Complete Case)
2.2 Existence of Extensions (Complete Case)
2.3 Extensions of Discrete Prime Divisors
2.4 Extensions in the General Case
2.5 Consequences; Exercises
Chapter 3. Local Fields
3.1 Newton's Method
3.2 Unramified Extensions
3.3 Totally Ramified Extensions
3.4 Tamely Ramified Extensions
3.5 Inertia Group
3.6 Ramification Groups
3.7 Different and Discriminant; Exercises
Chapter 4. Ordinary Arithmetic Fields
4.1 Axioms and Basic Properties
4.2 Ideals and Divisors
4.3 The Fundamental Theorem of OAFs
4.4 Dedekind Rings
4.5 Over-rings of O
4.6 Class Number
4.7 Mappings of Ideals
4.8 Different and Discriminant
4.9 Factoring Prime Ideals in an Extension Field
4.10 Hilbert Theory; Exercises
Chapter 5. Global Fields
5.1 Global Fields and the Product Formula
5.2 Adèles, Idèles, Divisors, and Ideals
5.3 Unit Theorem and Class Number
5.4 Class Number of an Algebraic Number Field
5.5 Topological Considerations
5.6 Relative Theory; Exercises
Chapter 6. Quadratic Fields
6.1 Integral Basis and Discriminant
6.2 Prime Ideals
6.3 Units
6.4 Class Number
6.5 The Local Situation
6.6 Norm Residue Symbol
Chapter 7. Cyclotomic Fields
7.1 Elementary Facts
7.2 Unramified Primes
7.3 Quadratic Reciprocity Law
7.4 Ramified Primes
7.5 Integral Basis and Discriminant
7.6 Units
7.7 Class Number
Symbols and Notation; Index