Fields and Galois theory - Howie, John M. (John Mackintosh) (Springer undergraduate mathematics series, 2006)
This book provides a gentle introduction to Galois theory suitable for third- and fourth-year undergraduates and beginning graduates. The approach is unashamedly unhistorical: it uses the language and techniques of abstract algebra to express complex arguments in contemporary terms. Thus the insolubility of the quintic by radicals is linked to the fact that the alternating group of degree 5 is simple - which is assuredly not the way Galois would have expressed the connection. Topics covered include rings and fields, integral domains and polynomials, field extensions and splitting fields, applications to geometry, finite fields, the Galois group, equations. Group theory features in many of the arguments, and is fully explained in the text. Clear and careful explanations are backed up with worked examples and more than 100 exercises, for which full solutions are provided.