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 Hubert 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 ? 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. - AppendixB 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. Language: English