The present volume 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 give 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.

INTRODUCTION

SECTION

1. Extension of homomorphisms

2. Algebras

3. Tensor products of vector spaces

4. Tensor product of algebras

CHAPTER I￡oFINITE 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. Normal bases

13. Finite fields

14. Regular representation, trace and norm

15. Galois cohomology

16. Composites of fields

CHAPTER II￡oGALOIS THEORY OF EQUATIONS

1. The Galois group of an equation

2. Pure equations

3. Galois' criterion for solvability by radicals

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