域和伽罗瓦理论(英文版)

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