Galois Theory, 5th Edition

Chapter 17Abstract Field Extensions and Galois Groups

DOI: 10.1201/9781003213949-17

Having defined rings and fields, and equipped ourselves with several methods for constructing them, we are now in a position to attack the general structure of an abstract field extension. Our previous work with subfields of $ℂ$ paves the way, and most of the effort goes into making minor changes to terminology and checking carefully that the underlying ideas generalise in the obvious manner.

We begin by extending the classification of simple extensions to general fields. Having done that, we assure ourselves that the theory of normal extensions, including their relation to splitting fields, carries over to the general case. A new issue, separability, comes into ...