Fields and Galois Theory

The pioneering work of Abel and Galois in the early nineteenth century demonstrated that the long-standing quest for a solution of quintic equations by radicals was fruitless: no formula can be found. The techniques they used were, in the end, more important than the resolution of a somewhat esoteric problem, for they were the genesis of modern abstract algebra.

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.

Texte du rabat

A modern and student-friendly introduction to this popular subject: it takes a more "natural" approach and develops the theory at a gentle pace with an emphasis on clear explanations

Features plenty of worked examples and exercises, complete with full solutions, to encourage independent study

Previous books by Howie in the SUMS series have attracted excellent reviews

Contenu

Rings and Fields.- Integral Domains and Polynomials.- Field Extensions.- Applications to Geometry.- Splitting Fields.- Finite Fields.- The Galois Group.- Equations and Groups.- Some Group Theory.- Groups and Equations.- Regular Polygons.- Solutions.