《域论(第2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，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。本书由罗曼著。

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