PREFACE

Summary

I wrote this book for myself.

I wanted to piece together carefully my own path through Galois Theory, a subject whose mathematical centrality and beauty I had often glimpsed, but one which I had never properly organized in my own mind. I wanted to start with simple, interesting questions and solve them as quickly and directly as possible. If related interesting questions arose along the way, I would deal with them too, but only if they seemed irresistible. I wanted to avoid generality for its own sake, and, as far as practicable; even generality that could only be appreciated in retrospect. Thus, I approached this project as an inquirer rather than as an expert, and I hope to share some of the sense of discovery and excitement I experienced. There is great mathematics here.

In particular, the book presents an exposition of those portions of classical field theory which are encountered in the solution of the famous geometric construction problems of antiquity and the problem of solving polynomial equations by radicals. Some time ago much of this material was covered in undergraduate courses in the ‘theory of equations’. Paradoxically, as the theory matured and became more elegant, it also moved higher into the curriculum, so that nowadays it is not uncommon for it first to be encountered on the graduate level.