《域论(第2版)(英文版)》是一部研究生水平的域论的入门书籍。每节后面都有不少练习，使得本书既是一本很好的教程，也是一本不错的参考书。本书从头开始阐述了域基本理论，如果具备本科生水平的抽象代数知识将对学习本书具有很大的帮助。本书是第二版，作者基于第一版及在运用第一版在教学过程中的经验，又将本书中的基本内容进行了改进。增加了新的练习和新的一章从历史展望角度讲述了Galois理论，通书不断涌现新话题，包括代数基本理论的证明、不可约情形的讨论、Zp上多项式因式分解的Berlekamp代数等。目次：基础；（第一部分）域扩展：多项式；域扩展；嵌入和可分性；代数独立性；（第二部分）Galois理论Ⅰ，历史回顾；Galois理论Ⅱ，理论；Galois理论Ⅲ，多项式的Galois群；域扩展作为向量空间；有限域Ⅰ，基本性质；有限域Ⅱ，附加性质；单位根；循环扩张；可解性扩张；（第三部分）二项式；二项式族。

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.