Galois Theory and Its Algebraic Background (2^nd Ed.)

Galois Theory, the theory of polynomial equations and their solutions, is one of the most fascinating and beautiful subjects of pure mathematics. Using group theory and field theory, it provides a complete answer to the problem of the solubility of polynomial equations by radicals: that is, determining when and how a polynomial equation can be solved by repeatedly extracting roots using elementary algebraic operations. This textbook contains a fully detailed account of Galois Theory and the algebra that it needs and is suitable both for those following a course of lectures and the independent reader (who is assumed to have no previous knowledge of Galois Theory). The second edition has been significantly revised and re-ordered; the first part develops the basic algebra that is needed, and the second a comprehensive account of Galois Theory. There are applications to ruler-and- compass constructions, and to the solution of classical mathematical problems of ancient times. There are new exercises throughout, and carefully-selected examples will help the reader develop a clear understanding of the mathematical theory.

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.

D. J. H. Garling is Emeritus Reader in Mathematical Analysis at the University of Cambridge and Fellow of St John's College, Cambridge. He has fifty years' experience of teaching undergraduate students and has written several books on mathematics, including Inequalities: A Journey into Linear Analysis (Cambridge, 2007) and A Course in Mathematical Analysis (Three volumes, Cambridge, 2013–2014).