Topics in Field TheoryBy
- G. Karpilovsky
This monograph gives a systematic account of certain important topics pertaining to field theory, including the central ideas, basic results and fundamental methods.Avoiding excessive technical detail, the book is intended for the student who has completed the equivalent of a standard first-year graduate algebra course. Thus it is assumed that the reader is familiar with basic ring-theoretic and group-theoretic concepts. A chapter on algebraic preliminaries is included, as well as a fairly large bibliography of works which are either directly relevant to the text or offer supplementary material of interest.
North-Holland Mathematics Studies
Published: February 1989
...well-written, should be useful to both students and specialists... makes a worthwhile contribution to the literature on field theory.
R. Massy, Mathematical Reviews
This book is intended for graduates familiar with basic ring- and group-theoretic concepts and gives a systematic account of important topics in field theory.
This is a very welcome monograph on certain topics in field theory written in a competent and rigorous style... The outstanding merit of the author lies in the ability to present the most interesting, important and deepest results in a coherent way. Thus the book will serve as a standard reference for all those who are interested in or are working on that part of algebra.
H. Mitsch, Monatshelfte fur Mathematik
- 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.