Galois Theory and Its Algebraic Background

This textbook contains a full account of Galois Theory and the algebra that it needs, with exercises, examples and applications.

Part I. The Algebraic Background: 1. Groups; 2. Integral domains; 3. Vector spaces and determinants; Part II. The Theory of Fields, and Galois Theory: 4. Field extensions; 5. Ruler and compass constructions; 6. Splitting fields; 7. Normal extensions; 8. Separability; 9. The fundamental theorem of Galois theory; 10. The discriminant; 11. Cyclotomic polynomials and cyclic extensions; 12. Solution by radicals; 13. Regular polygons; 14. Polynomials of low degree; 15. Finite fields; 16. Quintic polynomials; 17. Further theory; 18. The algebraic closure of a field; 19. Transcendental elements and algebraic independence; 20. Generic and symmetric polynomials; Appendix: the axiom of choice; Index.