Galois Theory, 5th Edition

Chapter 5Simple Extensions

DOI: 10.1201/9781003213949-5

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—one that is obtained by adjoining finitely many elements to K—can be obtained by a finite equence 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 ...