抽象代数讲义

INTRODUCTION

S ECTION

1. Extension of homomorphisms

2. Algebras

3. Tensor products of vector spaces

4. Tensor product of algebras

CHAPTER I: FINITE DIMENSIONAL EXTENSION FIELDS

1 Some vector spaces associated with mappings of fields

2. The Jacobson-Bourbaki correspondence

3. Dedekind independence theorem for isomorphisms of a field

4. Finite groups of automorphisms.

5. Splitting field of a polynomial

6. Multiple roots. Separable polynomials

7. The "fundamental theorem" of Galois theory

8. Normal extensions. Normal closures

9. Structure of algebraic extensions. Separability

10. Degrees of separability and inseparability. Structure of normal extensions

11. Primitive elements

12. Normalbases

13. Finitefields

14. Regular representation, trace and norm

15. Galois cohomology

16. Composites of fields

CHAPTER II: GALOIS THEORY OF EQUATIOIVS

1. The Galois group of an equation

2. Pureequations

3. Galois' criterion for solvability by radicals

4. The general equation of n-th degree

5. Equations with rational coefficients and symmetric group as Galoisgroup

CHAPTER Ⅲ: ABELIAN EXTENSlONS

1. Cyclotomic fields over the rationals

2. Characters of finite commutatiye groups

3. Kummer extensions

4. Witt rrectors

5. Abelian p-extensions

CHAPTER Ⅳ: STRUCTURE THEORY OF FIELDS

1 Algebraically closed fields

2. Infinite Galois theory

3. Transcendency basis

4. Luroth's theorem.

5. Linear disjointness and separating transcendency bases

6. Derivations

7. Derivations, separability and p-independence

8. Galois theory for purely inseparable extensions of exponert one

9. Higher derivations

10. Tensor products of fields

11. Free composites offields

CHAPTER V: VALUATION .THEORY

1. Realvaluations

2. Real valuations of the field of rational numbers

3. Real valuations of （x） which are trivial in

4. Completionofafield

5. Some properties of the field of p-adic numbers

6. Hensel'slemma

7. Construction of complete fields with given residue fields

8. Ordered groups and-valuations

9. Valuations, valuation rings, and places

10. Characterization of real non-archimedean valuations

11. Extension of homomorphisms and valuations

12. Application of the extension theorem: Hilbert Nullstellensatz

13. Application of the extension theorem: integral closure

……

CHAPTER VI: ARTIN-SCHREIER THEORY

Index

The present Vrolume completes the series of texts on algebra which the author began more than ten years ago. The account of field theory and Galois theory which we girre here is based on the notions and results of general algebra which appear in our first volume and on the more elementary parts of the second volume， dealing with linear algebra. The level of the present work .is roughly the same as that of Volume II.