Topics in Field Theory

Algebraic Preliminaries. Localization. Integral Extensions. Polynomial Rings. Unique Factorization Domains. Dedekind Domains. Separable Algebraic Extensions. Algebraic Closure, Splitting Fields and Normal Extensions. Separable Algebraic Extensions: Definitions and Elementary Properties. Separability, Linear Disjointness and Tensor Products. Norms, Traces and Discriminants of Separable Field Extensions. Transcendental Extensions. Abstract Dependence Relations. Transcendency Bases. Simple Transcendental Extensions. Separable Extensions. Weil's Order of Inseparability. Separability and Preservation of p-Independence. Perfect Ground Fields. Criteria for Separating Transcendency Bases. Separable Generation of Intermediate Field Extensions. The Steinitz Field Tower. Nonseparably Generated Fields over Maximal Perfect Subfields. Relatively Separated Extensions. Reliability and Relative Separability. Derivations. Definitions and Elementary Properties. Extensions of Derivations. Derivations, Separability and p-Independence. Restricted Subspace of DerE. Purely Inseparable Extensions. Preparatory Results for Splitting Theory. Splitting Theory. Chains of Splitting Fields and Complexity. Modular Extensions. Introduction and Preliminary Results. Pure Independence, Basic Subfields and Tensor Products of Simple Extensions. Ulm Invariants of Modular Extensions. Ulm Invariants and Group Algebras. Modular Closure and Modularly Perfect Fields. Galois Theory. Topological Prerequisites. Profinite Groups. Galois Extensions. Finite Fields, Roots of Unity and Cyclotomic Extensions. Finite Galois Theory. Infinite Galois Theory. Realizing Finite Groups as Galois Groups. Degrees of Sums in a Separable Field Extension. Galois Cohomology. Abelian Extensions. Witt Vectors. Cyclic Extensions. Abelian p-Extensions over Fields of Characteristic p. Kummer Theory. Character Groups of Infinite Abelian Extensions. Radical Extensions. Irreducibility of Binomials and Applications. Solvability of Galois Groups of Radical Extensions. Abelian Binomials. Normal Binomials. Some Additional Results. Cogalois Extensions. A Galois Correspondence for Radical Extensions. Duality of Lattices for Gal(E/F) and Cog(E/F). The Lattice of Intermediate Fields of Radical Extensions. Bibliography. Index.