This is a textbook on Galois theory. Galois theory has a well-deserved
reputation as one of the most beautiful subjects in mathematics. I was seduced by
its beauty into writing this book. I hope you will be seduced by its beauty in
reading it.
This book begins at the beginning. Indeed (and perhaps a little unusually
for a mathematics text), it begins with an informal introductory chapter,
Chapter 1. In this chapter we give a number of examples in Galois theory, even
before our terms have been properly defined. (Needless to say, even though
we proceed informally here, everything we say is absolutely correct.) These
examples are sort of an airport beacon, shining a clear light at our destination
as we navigate a course through the mathematical skies to get there.
Then we start with our proper development of the subject, in Chapter 2.
We assume no prior knowledge of field theory on the part of the reader. We
develop field theory, with our goal being the Fundamental Theorem of Galois
Theory (the FTGT). On the way, we consider extension fields, and deal with
the notions of normal, separable, and Galois extensions. Then, in the
penultimate section of this chapter, we reach our main goal, the FTGT.
Roughly speaking, the content of the FTGT is as follows: To every Galois
extension E of a field F we can associate its Galois group G = Gal(E/F). By
definition, G is the group of automorphisms of E that are the identity on F.
Then the FTGT establishes a one-to-one correspondence between fields B that
are intermediate between E and F, i.e., between fields B with F c B c E, and
subgroups of G. This connection allows us to use the techniques of group
theory to answer questions about fields that would otherwise be intractable.
(Indeed, historically Galois theory has been used to solve questions about fields
that were outstanding for centuries, and even for millenia. We will treat some
of these questions in this book.) In the final section of this chapter, we return
to the informal examples with which we started the book, as well as treat-