Field extension

A field extension [math]\displaystyle{ K }[/math] is a field containing a given field [math]\displaystyle{ k }[/math] as a subfield.

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

[math]\displaystyle{ K }[/math] is sometimes called an overfield of the field [math]\displaystyle{ k }[/math].

Algebraic over [math]\displaystyle{ k }[/math]

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