Finitely generated field extension

Let $E/F$ be a field extension . The extension $E/F$ is finitely generated (as a field extension) if there exist elements $\alpha_1,\dots,\alpha_n\in E$ such that

where $F(\alpha_1,\dots,\alpha_n)$ denotes the smallest subfield of $E$ containing $F$ and all $\alpha_i$. Equivalently, $E$ is generated as a field by a finite subset of $E$ over $F$.

The special case $n=1$ is exactly a simple extension . Finitely generated extensions include both finite extensions (finite degree ) and many transcendental extensions; for instance, adjoining one transcendental element already gives a finitely generated but typically infinite-degree extension.

When $[E:F]<\infty$, the extension is automatically finitely generated, and in many important cases it is even simple by the primitive element theorem .

Examples

1. $\mathbb{Q}(\sqrt2,\sqrt3)/\mathbb{Q}$ is finitely generated: $E=\mathbb{Q}(\sqrt2,\sqrt3)$ with two generators, and in fact it is an algebraic extension .
2. $\mathbb{Q}(t)/\mathbb{Q}$ is finitely generated (one generator $t$) but is a transcendental extension of infinite degree.
3. $\mathbb{Q}(t,\sqrt{t})/\mathbb{Q}$ is finitely generated: it is generated by $t$ and $\sqrt{t}$. It is transcendental over $\mathbb{Q}$ (because it contains $t$), but the intermediate extension $\mathbb{Q}(t,\sqrt{t})/\mathbb{Q}(t)$ is algebraic of degree $2$ since $\sqrt{t}$ satisfies $x^2-t=0$.