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 -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 -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 -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.

Ivan Fesenko