نظریه میدان‌های جابجایی (ترجمه‌های مونوگرافی‌های ریاضی)

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. 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

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.