Field Arithmetic

The book explores algebraic tools to study fields and related algorithmic problems, covering foundational material on infinite Galois theory, profinite groups, algebraic function fields, and plane curves. It provides complete and elementary proofs of the Chebotarev density theorem and the Riemann hypothesis for function fields, as well as material on ultraproducts, decision procedures, the elementary theory of algebraically closed fields, undecidability and nonstandard model theory. It also focuses on pseudo algebraically closed (PAC) fields, related structures, and associated decidability and undecidability results. The fourth edition substantially extends, updates, and clarifies the previous editions, with a new chapter on Hilbertian subfields of Galois extensions and an appendix presenting open research problems.

Format: Hardback
Length: 827 pages
Publication date: 13 June 2023
Publisher: Springer International Publishing AG

This book employs algebraic techniques to investigate the fundamental characteristics of field classes and related algorithmic challenges. The initial section delves into essential concepts, including infinite Galois theory, profinite groups, algebraic function fields in one variable, and plane curves. It offers comprehensive and elementary proofs of the Chebotarev density theorem and the Riemann hypothesis for function fields, along with discussions on ultraproducts, decision procedures, the elementary theory of algebraically closed fields, undecidability, and nonstandard model theory, including a nonstandard proof of Hilbert's irreducibility theorem. The focus subsequently shifts to the study of pseudo algebraically closed (PAC) fields, their associated structures, and the associated decidability and undecidability results. PAC fields, which are fields K with the property that every absolutely irreducible variety over K has a rational point, initially emerged in the elementary theory of finite fields and possess profound connections with number theory. This fourth edition significantly expands, updates, and clarifies the previous editions of this renowned book and includes a new chapter on Hilbertian subfields of Galois extensions. Nearly every chapter concludes with a set of exercises and bibliographical notes. An appendix presents a selection of open research problems.

This comprehensive and self-contained text, drawing from a diverse literature at the intersection of logic and arithmetic, can serve as a textbook for graduate courses and as a valuable reference for seasoned researchers.

Dimension: 235 x 155 (mm)
ISBN-13: 9783031280191
Edition number: 4th ed. 2023