Central Extensions, Galois Groups, and Ideal Class Groups of Number Fields (eBook)

These notes deal with a set of interrelated problems and results in algebraic number theory, in which there has been renewed activity in recent years. The underlying tool is the theory of the central extensions and, in most general terms, the underlying aim is to use class field theoretic methods to reach beyond Abelian extensions. One purpose of this book is to give an introductory survey, assuming the basic theorems of class field theory as mostly recalled in section 1 and giving a central role to the Tate cohomology groups $/hat H{}^{-1}$. The principal aim is, however, to use the general theory as developed here, together with the special features of class field theory over $/mathbf Q$, to derive some rather strong theorems of a very concrete nature, with $/mathbf Q$ as base field. The specialization of the theory of central extensions to the base field $/mathbf Q$ is shown to derive from an underlying principle of wide applicability. The author describes certain non-Abelian Galois groups over the rational field and their inertia subgroups, and uses this description to gain information on ideal class groups of absolutely Abelian fields, all in entirely rational terms. Precise and explicit arithmetic results are obtained, reaching far beyond anything available in the general theory. The theory of the genus field, which is needed as background as well as being of independent interest, is presented in section 2. In section 3, the theory of central extension is developed. The special features over ${/mathbf Q}$ are pointed out throughout. Section 4 deals with Galois groups, and applications to class groups are considered in section 5. Finally, section 6 contains some remarks on the history and literature, but no completeness is attempted.