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.

D. J. H. Garling (Author)

9781108969086, Cambridge University Press

Paperback / softback, published 22 July 2021

204 pages
22.9 x 15.1 x 1.8 cm, 0.36 kg

'Garling's book presents Galois theory in a style which is at once readable and compact. The necessary prerequisites are developed in the early chapters only to the extent that they are needed later. The proofs of the lemmas and main theorems are presented in as concrete a manner as possible, without unnecessary abstraction. Yet they seem remarkably short, without the difficulties being glossed over. In fact the approach throughout the book is down-to-earth and concrete … I can heartily recommend this book as an undergraduate text.' Bulletin of the London Mathematical Society

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.

Subject Areas: Algebra [PBF]