
Chapter 5
Simple Extensions
The basic building block of field theory is the simple field extension. Here one new
element α is adjoined to a given subfield K of C, along with all rational expressions in
that element over K. Any finitely generated extension can be obtained by a sequence
of simple extensions, so the structure of a simple extension provides vital information
about all of the extensions that we shall encounter.
We first classify simple extensions into two very different kinds: transcendental
and algebraic. If the new element α satisfies a polynomial equation over K, then
the extension is algebraic; if not, it is transcendental. Up to isomorphism, ...