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Local Fields and Their Extensions: Second Edition
I. B. Fesenko, University of Nottingham, England, and S. V. Vostokov, St. Petersburg University, Russia

Translations of Mathematical Monographs
2002; 345 pp; hardcover
Volume: 121
ISBN-10: 0-8218-3259-X
ISBN-13: 978-0-8218-3259-2
List Price: US$129
Member Price: US$103.20
Order Code: MMONO/121.R
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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.


Graduate students and research mathematicians interested in local number theory and its applications in arithmetic algebraic geometry.


Reviews of the Previous Edition:

"It is remarkable to see just how far the subject has developed since 1968 ... contains an absolute wealth of material ... this approach is a real success ... results are obtained with a minimum of fuss, so that the story unfolds rather quickly and holds the reader's interest ... a copious supply of well-structured exercises ... most certainly a valuable addition to the literature ... carefully written and well-presented state of the art account of local fields, which contains much ... of interest to the expert and non-expert alike ... its appeal should go well beyond the usual public of number theorists."

-- Bulletin of the London Mathematical Society

"Well written ... A big amount of exercises contribute to the attraction of this highly original book."

-- Zentralblatt MATH

Table of Contents

  • Complete discrete valuation fields
  • Extensions of discrete valuation fields
  • The norm map
  • Local class field theory I
  • Local class field theory II
  • The group of units of local number fields
  • Explicit formulas for the Hilbert symbol
  • Explicit formulas for the Hilbert pairing on formal groups
  • The Milnor \(K\)-groups of a local field
  • Bibliography
  • List of notations
  • Index
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