●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

域是有理數集合、實數集合、復數集合的抽像模型，因此在整個數學科學中處於基礎地位。Galois是最早提出有限域觀點的人，他對於抽像域理論的誕生至關重要 本書把抽像域論一分為二，首先講代數擴張及其在代數域論上的應用，其次介紹超越擴張及其在代數函數論及代數幾何上的應用，中間還插入經典的Galois理論，使讀者對於實際背景有比較清楚的認識。