Fourier Analysis on Number Fields

The general aim of this book is to provide a modern approach to number theory through a blending of complementary algebraic and analytic perspectives, emphasizing harmonic analysis on topological groups. The more particular goal is to cover John Tate's visionary thesis, giving virtually all of the necessary analytic details and topological preliminaries--technical prerequisites that are often foreign to the typical, more algebraically inclinded number theorist.

Zusammenfassung
A modern approach to number theory through a blending of complementary algebraic and analytic perspectives, emphasising harmonic analysis on topological groups. In fact, the subject matter at hand is germane not only to budding number theorists, but also to students of harmonic analysis or the representation theory of Lie groups.

Inhalt
1 Topological Groups.- 2 Some Representation Theory.- 3 Duality for Locally Compact Abelian Groups.- 4 The Structure of Arithmetic Fields.- 5 Adeles, Ideles, and the Class Groups.- 6 A Quick Tour of Class Field Theory.- 7 Tate's Thesis and Applications.- Appendices.- Appendix A: Normed Linear Spaces.- A.1 Finite-Dimensional Normed Linear Spaces.- A.2 The Weak Topology.- A.3 The Weak-Star Topology.- Appendix B: Dedekind Domains.- B.1 Basic Properties.- B.2 Extensions of Dedekind Domains.- References.