域是有理数集合、实数集合、复数集合的抽象模型。

本书为英文版，把抽象域论一分为二，首先讲代数扩张及其在代数域论上的应用，其次介绍超越扩张及其在代数函数论及代数几何上的应用，中间还插入经典的Galois理论，使读者对于实际背景有比较清楚的认识。

域是有理数集合、实数集合、复数集合的抽象模型，因此在整个数学科学中处于基础地位。Galois是最早提出有限域观点的人，他对于抽象域理论的诞生至关重要。 本书把抽象域论一分为二，首先讲代数扩张及其在代数域论上的应用，其次介绍超越扩张及其在代数函数论及代数几何上的应用，中间还插入经典的Galois理论，使读者对于实际背景有比较清楚的认识。

Preface

Notes to the Reader

List of Symbols

I Galois Theory

1 Field Extensions

2 Automorphisms

3 Normal Extensions

4 Separable and Inseparable Extensions

5 The Fundamental Theorem of Galois Theory

II Some Galois Extensions

6 Finite Fields

7 Cyclotomic Extensions

8 Norms and Traces

9 Cyclic Extensions

10 Hilbert Theorem 90 and Group Cohomology.

11 Kummer Extensions

III Applications of Galois Theory

12 Discriminants

13 Polynomials of Degree 3 and 4

14 The Transcendence of 7r and e

15 Ruler and Compass Constructions

16 Solvability by Radicals

IV Infinite Algebraic Extensions

17 Infinite Galois Extensions

18 Some Infinite Galois Extensions

V Transcendental Extensions

19 Transcendence Bases

20 Linear Disjointness

21 Algebraic Varieties

22 Algebraic Function Fields

23 Derivations and Differentials

Appendix A Ring Theory

1 Prime and Maximal Ideals

2 Unique Factorization Domains

3 Polynomials over a Field

4 Factorization in Polynomial Rings

5 Irreducibility Tests

Appendix B Set Theory

1 Zorn's Lemma

2 Cardinality and Cardinal Arithmetic

Appendix C Group Theory

1 Fundamentals of Finite Groups

2 The Sylow Theorems

3 Solvable Groups

4 Profinite Groups

Appendix D Vector Spaces

1 Bases and Dimension

2 Linear Transformations

3 Systems of Linear Equations and Determinants

4 Tensor Products

Appendix E Topology

1 Topological Spaces

2 Topological Properties

References

Index