域論

《域論(第2版)(英文版)》是一部研究生水平的域論的入門書籍。每節後面都有不少練習，使得本書既是一本很好的教程，也是一本不錯的參考書。本書從頭開始闡述了域基本理論，如果具備本科生水平的抽象代數知識將對學習本書具有很大的幫助。本書是第二版，作者基於第一版及在運用第一版在教學過程中的經驗，又將本書中的基本內容進行了改進。增加了新的練習和新的一章從歷史展望角度講述了Galois理論，通書不斷湧現新話題，包括代數基本理論的證明、不可約情形的討論、Zp上多項式因式分解的Berlekamp代數等。目次：基礎；（第一部分）域擴展：多項式；域擴展；嵌入和可分性；代數獨立性；（第二部分）Galois理論Ⅰ，歷史回顧；Galois理論Ⅱ，理論；Galois理論Ⅲ，多項式的Galois群；域擴展作為向量空間；有限域Ⅰ，基本性質；有限域Ⅱ，附加性質；單位根；循環擴張；可解性擴張；（第三部分）二項式；二項式族。

preface

contents

0 preliminaries

0.1 lattices

0.2 groups

0.3 the symmetric group

0.4 rings

0.5 integral domains

0.6 unique factorization domains

0.7 principal ideal domains

0.8 euclidean domains

0.9 tensor products

exercises

part i-field extensions

1 polynomials

1.1 polynomials over a ring

1.2 primitive polynomials and irreducibility

1.3 the division algorithm and its consequences

1.4 splitting fields

.1.5 the minimal polynomial

1.6 multiple roots

1.7 testing for irreducibility

exercises

2 field extensions

2.1 the lattice of subfields of a field

2.2 types of field extensions

2.3 finitely generated extensions

2.4 simple extensions

2.5 finite extensions

2.6 algebraic extensions

2.7 algebraic closures

2.8 embeddings and their extensions.

2.9 splitting fields and normal extensions

exercises

3 embeddings and separability

3.1 recap and a useful lemma

3.2 the number of extensions: separable degree

3.3 separable extensions

3.4 perfect fields

3.5 pure inseparability

3.6 separable and purely inseparable closures

exercises

4 algebraic independence

4.1 dependence relations

4.2 algebraic dependence

4.3 transcendence bases

4.4 simple transcendental extensions

exercises

part ii——-galois theory

5 galois theory i: an historical perspective

5.1 the quadratic equation

5.2 the cubic and quartic equations

5.3 higher-degree equations

5.4 newton's contribution: symmetric polynomials

5.5 vandermonde

5.6 lagrange

5.7 gauss

5.8 back to lagrange

5.9 galois

5.10 a very brief look at the life of galois

6 galois theory i1: the theory

6.1 galois connections

6.2 the galois correspondence

6.3 who's closed?

6.4 normal subgroups and normal extensions

6.5 more on galois groups

6.6 abelian and cyclic extensions

*6.7 linear disjointness

exercises

7 galois theory iii: the galois group of a polynomial

7.1 the galois group of a polynomial

7.2 symmetric polynomials

7.3 the fundamental theorem of algebra.

7.4 the discriminant of a polynomial

7.5 the galois groups of some small-degree polynomials

exercises

8 a field extension as a vector space

8.1 the norm and the trace

*8.2 characterizing bases

*8.3 the normal basis theorem

exercises

9 finite fields i: basic properties

9.1 finite fields redux

9.2 finite fields as splitting fields

9.3 the subfields of a finite field.

9.4 the multiplicative structure of a finite field

9.5 the galois group of a finite field

9.6 irreducible polynomials over finite fields

*9.7 normal bases

*9.8 the algebraic closure of a finite field

exercises

10 finite fields i1: additional properties

10.1 finite field arithmetic

10.2 the number of irreducible polynomials

10.3 polynomial functions

10.4 linearized polynomials

exercises

11 the roots of unity

11.1 roots of unity

11.2 cyclotomic extensions

11.3 normal bases and roots of unity

11.4 wedderburn's theorem

11.5 realizing groups as galois groups

exercises

12 cyclic extensions

12.1 cyclic extensions

12.2 extensions of degree char（f）

exercises

13 solvable extensions

13.1 solvable groups

13.2 solvable extensions

13.3 radical extensions

13.4 solvability by radicals

13.5 solvable equivalent to solvable by radicals

13.6 natural and accessory irrationalities

13.7 polynomial equations

exercises

part iii——the theory of binomials

14 binomials

14.1 irreducibility

14.2 the galois group of a binomial

14.3 the independence of irrational numbers

exercises

15 families of binomials

15.1 the splitting field

15.2 dual groups and pairings

15.3 kummer theory

exercises

appendix: mobius inversion

partially ordered sets

the incidence algebra of a partially ordered set

classical mobius inversion

multiplicative version of m6bius inversion

references

index

This book presents the basic theory of fields, starting more or less from the beginning. It is suitable for a graduate course in field theory, or independent study. The reader is expected to have taken an undergraduate course in abstract algebra, not so much for the material it contains but in order to gain a certain level of mathematical maturity.

The book begins with a preliminary chapter (Chapter 0), which is designed to be quickly scanned or skipped and used as a reference if needed. The remainder of the book is divided into three parts.

Part 1, entitled Field Extensions, begins with a chapter on polynomials. Chapter 2 is devoted to various types of field extensions, including finite, finitely generated, algebraic and normal. Chapter 3 takes a close look at the issue of separability. In my classes, I generally cover only Sections 3.1 to 3.4 (on perfect fields). Chapter 4 is devoted to algebraic independence, starting with the general notion of a dependence relation and concluding with Luroth's theorem on intermediate fields of a simple transcendental extension.

Part 2 of the book is entitled Galois Theory. Chapter 5 examines Galois theory from an historical perspective, discussing the contributions from Lagrange,Vandermonde, Gauss, Newton, and others that led to the development of the theory. I have also included a very brief look at the very brief life of Galois himself.

Chapter 6 begins with the notion of a Galois correspondence between two partially ordered sets, and then specializes to the Galois correspondence of a field extension, concluding with a brief discussion of the Krull topology. In Chapter 7, we discuss the Galois theory of equations. In Chapter 8, we view a field extension E of F as a vector space over F.

Chapter 9 and Chapter 10 are devoted to finite fields, although this material can be omitted in order to reach the topic of solvability by radicals more quickly.Mobius inversion is used in a few places, so an appendix has been included on this subject.

Part 3 of the book is entitled The Theory of Binomials. Chapter 11 covers the roots of unity and Wedderbum's theorem on finite division rings. We also briefly discuss the question of whether a given group is the Galois group of a field extension. In Chapter 12, we characterize cyclic extensions and splitting fields of binomials when the base field contains appropriate roots of unity.Chapter 13 is devoted to the question of solvability of a polynomial equation by radicals. (This chapter might make a convenient ending place in a graduate course.) In Chapter 14, we determine conditions that characterize the irreducibility of a binomial and describe the Galois group of a binomial. Chapter 15 briefly describes the theory of families of binomials--the so-called Kummer theory.

Sections marked with an asterisk may be skipped without loss of continuity. hanges for the Second Edition

Let me begin by thanking the readers of the first edition for their many helpful comments and suggestions.

For the second edition, I have gone over the entire book, and rewritten most of it, including the exercises. I believe the book has benefited significantly from a class testing at the beginning graduate level and at a more advanced graduate level.

I have also rearranged the chapters on separability and algebraic independence,feeling that the former is more important when time is of the essence. In my course, I generally touch only very lightly (or skip altogether) the chapter on algebraic independence, simply because of time constraints.

As mentioned earlier, as several readers have requested, 1 have added a chapter on Galois theory from an historical perspective.

A few additional topics are sprinkled throughout, such as a proof of the Fundamental Theorem of Algebra, a discussion of casus irreducibilis,Berlekamp's algorithm for factoring polynomials over Zp and natural and accessory irrationalities.

Thanks

I would like to thank my students Phong Le, Sunil Chetty, Timothy Choi and Josh Chan, who attended lectures on essentially the entire book and offeredmany helpful suggestions. I would also like to thank my editor, Mark Spencer,who puts up with my many requests and is most amiable.