# Local fields and their extensions / I. B. Fesenko, S. V. Vostokov

Type de document : MonographieCollection : Translations of mathematical monographs, 121Langue : anglais.Pays: Etats Unis.Mention d'édition: 2nd editionÉditeur : Providence : American Mathematical Society, 2002Description : 1 vol. (XI-345 p.) ; 27 cmISBN: 9780821832592.ISSN: 0065-9282.Bibliographie : Bibliogr. p. 319-339. Index.Sujet MSC : 11Sxx, Number theory - Algebraic number theory: local and p-adic fields11S31, Algebraic number theory: local and p-adic fields, Class field theory; p-adic formal groups

11S70, Algebraic number theory: local and p-adic fields, K-theory of local fields

11S15, Algebraic number theory: local and p-adic fields, Ramification and extension theory

11S20, Algebraic number theory: local and p-adic fields, Galois theoryEn-ligne : Zentralblatt | MathSciNet | AMS Item type: Monographie

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Bibliogr. p. 319-339. Index

This book offers a modern exposition of the arithmetical properties of local fields using explicit and constructive tools and methods. It has been ten years since the publication of the first edition, and, according to Mathematical Reviews, 1,000 papers on local fields have been published during that period. This edition incorporates improvements to the first edition, with 60 additional pages reflecting several aspects of the developments in local number theory.

The volume consists of four parts: elementary properties of local fields, class field theory for various types of local fields and generalizations, explicit formulas for the Hilbert pairing, and Milnor K-groups of fields and of local fields. The first three parts essentially simplify, revise, and update the first edition. The book includes the following recent topics: Fontaine-Wintenberger theory of arithmetically profinite extensions and fields of norms, explicit noncohomological approach to the reciprocity map with a review of all other approaches to local class field theory, Fesenko's p-class field theory for local fields with perfect residue field, simplified updated presentation of Vostokov's explicit formulas for the Hilbert norm residue symbol, and Milnor K-groups of local fields. Numerous exercises introduce the reader to other important recent results in local number theory, and an extensive bibliography provides a guide to related areas.

The book is designed for graduate students and research mathematicians interested in local number theory and its applications in arithmetic algebraic geometry. (source : AMS)

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