Field extension

A field extension ${\displaystyle K}$ is a field containing a given field ${\displaystyle k}$ as a subfield.^[1]

The notation ${\displaystyle K/k}$ means that ${\displaystyle K}$ is an extension of the field ${\displaystyle k}$.

${\displaystyle K}$ is sometimes called an overfield of the field ${\displaystyle k}$.

Algebraic over k {\displaystyle k}

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