Local fields and their extensions

This book offers a modern presentation of local fields whose spectacular development was initiated almost one hundred years ago by K. Hensel. The volume consists of nine chapters divided into four parts: arithmetic 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. This second edition contains about sixty additional pages reflecting several aspects of the developments in local number theory in the last ten years. Exercises point to many other paths for exploration. The book is designed for graduate students and research mathematicians interested in local number theory and its applications in arithmetic algebraic geometry.