
Chapter 11
Field Automorphisms
The theme of this chapter is the construction of automorphisms to given speci-
fications. We begin with a generalisation of a K-automorphism, known as a K-
monomorphism. For normal extensions we shall use K-monomorphisms to build up
K-automorphisms. Using this technique, we can calculate the order of the Galois
group of any finite normal extension, which combines with the result of Chapter 10
to give a crucial part of the fundamental theorem of Chapter 12.
We also introduce the concept of a normal closure of a finite extension. This
useful device enables us to steer around some of the technical obstructions caused by
non-normal ...