Definition 17.6.1 (Field extension). A field extension is a field containing as a subfield. The degree is the dimension of as a -vector space. An extension is finite if .
Proposition 17.6.1 (Tower law). If are fields, then .
Proof. If is an -basis for and is a -basis for , then is a -basis for .∎
Definition 17.6.2 (Algebraic and transcendental elements). Let be a field extension and .
is algebraic over if there exists a nonzero polynomial with .
is transcendental over if no such polynomial exists.
The minimal polynomial of an algebraic element is the unique monic polynomial of least degree with . It is irreducible and generates .
Proposition 17.6.2 (Simple algebraic extensions). If is algebraic over with minimal polynomial of degree , then is a field extension of degree , with basis over .
Example 17.6.1. The minimal polynomial of over is . So has degree 2 over , with basis . The minimal polynomial of over is (irreducible by Eisenstein at ), giving .