Fields and Galois Theory

Description

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. Front Matter....Pages i-x Rings and Fields....Pages 1-24 Integral Domains and Polynomials....Pages 25-49 Field Extensions....Pages 51-69 Applications to Geometry....Pages 71-78 Splitting Fields....Pages 79-84 Finite Fields....Pages 85-90 The Galois Group....Pages 91-126 Equations and Groups....Pages 127-147 Some Group Theory....Pages 149-168 Groups and Equations....Pages 169-181 Regular Polygons....Pages 183-192 Solutions....Pages 193-217 Back Matter....Pages 219-225