Finitely generated field extension

Definition. A field extension $L/K$ is finitely generated if there exist elements $\alpha_1,\dots,\alpha_r\in L$ such that

the smallest subfield of $L$ containing $K$ and all $\alpha_i$.
Equivalently, $L$ is obtained from $K$ by adjoining finitely many elements, possibly algebraic and/or transcendental.

A simple extension is the special case $r=1$.

See also. algebraic extension , transcendental extension .

Examples.

1. $\mathbb{Q}(\sqrt2,\sqrt3)/\mathbb{Q}$ is finitely generated (and finite).
2. $\mathbb{Q}(t,\sqrt{t})/\mathbb{Q}$ is finitely generated: first adjoin transcendental $t$, then adjoin $\sqrt t$.
3. The rational function field $K(x,y)=K(x,y)$ is finitely generated over $K$ by $\{x,y\}$ but has infinite degree.