Chapter 11Field Automorphisms
The theme of this chapter is the construction of automorphisms to given specifications. 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 extensions.
11.1 K-Monomorphisms
We begin by generalising ...
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