Field extension

A field extension $K$ is a field containing a given field $k$ as a subfield.

The notation $K/k$ means that $K$ is an extension of the field $k$.

$K$ is sometimes called an overfield of the field $k$.

Algebraic over $k$

An element of the field $K$ is called algebraic over $k$ if it satisfies some algebraic equation with coefficients in $k$, and transcendental otherwise.