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.
Author(s): Masayoshi Nagata
Series: Translations of Mathematical Monographs, Vol. 125
Publisher: American Mathematical Society
Year: 1993
Language: English
Pages: C+xvi+249+B
Cover
Translations of Mathematical Monographs 125
S Title
Theory of Commutative Fields
Copyright ©1993 by the American Mathematical Society
Copyright © 1967, 1985 by Masayoshi Nagata and Shokabo
ISBN 0-8218-4572-1
QA247.N2613 1993 512'.74-dc2O
LCCN 93-6503 CIP
Contents
Preface to the English Edition
Preface to the New Japanese Edition
Preface to the Original Japanese Edition
CHAPTER 0 Prerequisites from Set Theory
§0. Basic symbols
§1. Mappings
§2. Ordered sets
§3. Partitions and equivalence relations
CHAPTER I Groups, Rings, and Fields
§1. Groups
§2. Normal subgroups and homomorphisms
§3. Rings and fields
§4. Integral domains and prime ideals
§5. Polynomial rings
§6. Unique factorization
§7. Modules
§8. Symmetric forms and alternating forms
Exercises
CHAPTER II Algebraic Extensions of Finite Degrees
§1. Basic notions
§2. Splitting fields
§3. Separability and inseparability
§4. Multiplicative groups of finite fields
§5. Simple extensions
§6. Normal extensions
§7. Invariants of a finite group
§8. The fundamental theorem of Galois
§9. Roots of unity and cyclic extensions
§ 10. Solvability of algebraic equations
§11. Construction problems
§12. Algebraically closed fields
Appendix 1
Appendix 2
Exercises
CHAPTER III Transcendental Extensions
§1. Transcendence bases
§2. Tensor products over a field
§3. Derivations
§4. Separable extensions
§5. Regular extensions
§6. Noetherian rings
§7. Integral extensions and prime ideals
§8. The normalization theorem for polynomial rings
§9. Integral closures
§10. Algebraic varieties
§11. The C,-conditions
§12. The theorem of Luroth
Appendix. A theorem on valuation rings and its applications
Exercises
CHAPTER IV Valuations
§1. Multiplicative valuations
§2. Valuations of the rational number field
§3. Topology
§4. Topological groups and topological fields
§5. Completions
§6. Archimedean valuations and absolute values
§7. Additive valuations and valuation rings
§8. Approximation theorems
§9. Prolongations of a valuation
§10. The product formula
§11. Hensel's lemma
Exercises
CHAPTER V Formally Real Fields
§1. Ordered fields, formally real fields, and real closed fields
§2. Real closures
§3. Hilbert's 17th Problem
§4. A valuation corresponding to an order
Exercises
CHAPTER VI Galois Theory of Algebraic Extensions of Infinite Degree
§1. Topology on a Galois group
§2. The fundamental theorem of Galois
§3. Splitting fields, inertia fields, and ramification fields
§4. Algebraic equations of high degrees
Exercises
Answers and Hints
CHAPTER I
EXERCISES
CHAPTER II. EXERCISES
CHAPTER III. EXERCISES
CHAPTER IV. EXERCISES
CHAPTER V. EXERCISES
CHAPTER VI. EXERCISES
Index of Symbols
Subject Index
Back Cover
