**Translations of Mathematical Monographs**

2002;
345 pp;
Hardcover

MSC: Primary 11;

Print ISBN: 978-0-8218-3259-2

Product Code: MMONO/121.R

List Price: $129.00

AMS Member Price: $103.20

MAA Member Price: $116.10

**Electronic ISBN: 978-1-4704-4664-2
Product Code: MMONO/121.R.E**

List Price: $129.00

AMS Member Price: $103.20

MAA Member Price: $116.10

# Local Fields and Their Extensions: Second Edition

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*I. B. Fesenko; S. V. Vostokov*

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

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.

#### Readership

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

#### Reviews & Endorsements

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

# Table of Contents

## Local Fields and Their Extensions: Second Edition

- Cover Cover11
- Title page iii4
- Contents v6
- Introduction to the second edition vii8
- Foreword to the first edition by I.R. Shafarevich xi12
- Chapter I. Complete discrete valuation fields 114
- Chapter II. Extensions of discrete valuation fields 3548
- Chapter III. The norm map 6780
- Chapter IV. Local class field theory I 111124
- Chapter V. Local class field theory II 171184
- Chapter VI. The group of units of local number fields 207220
- Chapter VII. Explicit formulas for the Hilbert symbol 235248
- Chapter VIII. Explicit formulas for the Hilbert pairings on formal groups 267280
- Chapter IX. The Milnor 𝐾-groups of a local field 283296
- Bibliography 319332
- List of Notations 341354
- Index 343356
- Back Cover Back Cover1362