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。