Absolute values and their completions such as the p-adic number fields play an important role in number theory. Krull's generalization of absolute values to valuations made applications in other branches of mathematics, such as algebraic geometry, possible. In valuation theory, the notion of a completion has to be replaced by that of "Henselization". This book develops the theory of valuations as well as of Henselizations, based on the knowledge acquired in a standard graduate course in algebra.

Includes new chapter on valuation rings, definable in the language of fields only Contains in addition a Galois theoretic characterization of the field of p-adic numbers Written by experts Includes supplementary material: sn.pub/extras

Inhalt
Absolute Values.- Valuations.- Extension of Valuations.- Henselian Fields.- Structure Theory.- Applications of Valuation Theory.