**Translations of Mathematical Monographs**

1993;
249 pp;
Hardcover

MSC: Primary 12;

Print ISBN: 978-0-8218-4572-1

Product Code: MMONO/125

List Price: $110.00

AMS Member Price: $88.00

MAA Member Price: $99.00

**Electronic ISBN: 978-1-4704-4533-1
Product Code: MMONO/125.E**

List Price: $110.00

AMS Member Price: $88.00

MAA Member Price: $99.00

# Theory of Commutative Fields

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*Masayoshi Nagata*

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 prerequisites 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.

#### Readership

Graduate students and research mathematicians.

#### Reviews & Endorsements

The book … gives an interesting and very clear introduction to commutative algebra.

-- Mathematical Reviews

Very much self-contained.

-- Zentralblatt MATH

Students wanting a thorough introduction into field theory will profit by reading it, all the more as a bunch of exercises is added together with answers and hints.

-- Monatshefte für Mathematik

#### Table of Contents

# Table of Contents

## Theory of Commutative Fields

- Cover Cover11
- Title page v6
- Contents vii8
- Preface to the English Edition xi12
- Preface to the new Japanese Edition xiii14
- Preface to the original Japanese Edition xv16
- Chapter 0. Prerequisites from set theory 118
- Chapter I. Groups, rings, and fields 724
- Chapter II. Algebraic extensions of finite degrees 4360
- Chapter III. Transcendental extensions 8198
- Chapter IV. Valuations 135152
- Chapter V. Formally real fields 191208
- Chapter VI. Galois theory of algebraic extensions of infinite degree 215232
- Answers and hints 227244
- Index of symbols 243260
- Subject index 245262
- Back Cover Back Cover1271