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
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