Fields and Galois Theory

  • John M. Howie

Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Table of contents

  1. Front Matter
    Pages i-x
  2. Pages 1-24
  3. Pages 51-69
  4. Pages 79-84
  5. Pages 85-90
  6. Pages 91-126
  7. Pages 127-147
  8. Pages 149-168
  9. Pages 169-181
  10. Pages 183-192
  11. Pages 193-217
  12. Back Matter
    Pages 219-225

About this book


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


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.


Abstract algebra Field theory Galois theory Group theory Polynomials algebra finite field

Authors and affiliations

  • John M. Howie
    • 1
  1. 1.School of Mathematics and StatisticsUniversity of St AndrewsNorth Haugh St Andrews FifeUK

Bibliographic information