The theory of commutative fields is a fundamental area of mathematics, particularly in number theory, algebra, and algebraic geometry. However, few books provide sufficient treatment of this topic. This book is a translation of the 1985 updated edition of Nagata's 1967 book; both editions originally appeared in Japanese. Nagata provides an introduction to commutative fields that is useful to those studying the topic for the first time as well as to those wishing a reference book. The book presents, with as few prerequisities as possible, all of the important and fundamental results on commutative fields. Each chapter ends with exercises, making the book suitable as a textbook for graduate courses or for independent study.
Prerequisites from set theory Groups, rings, and fields Algebraic extensions of finite degrees Transcendental extensions Valuations Formally real fields Galois theory of algebraic extensions of infinite degree Answers and hints Index of symbols Subject index.