
Chapter 17
Abstract Field Extensions
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. ...