Algebraic extension

A field extension $E/F$ is called an algebraic extension if every element $\alpha\in E$ is an algebraic element over $F$; that is, for each $\alpha\in E$ there exists a nonzero polynomial $f(x)\in F[x]$ with $f(\alpha)=0$.

Equivalently, $E/F$ is algebraic iff for every $\alpha\in E$, the simple subextension $F(\alpha)/F$ (see simple extension ) has finite degree . In particular, every finite extension $[E:F]<\infty$ is algebraic.

Algebraic extensions are the setting for splitting fields and Galois theory: for example, a Galois extension is an algebraic extension satisfying additional normality and separability conditions.

Examples

1. $\mathbb{Q}(\sqrt2,\sqrt3)/\mathbb{Q}$ is algebraic because $\sqrt2$ and $\sqrt3$ are algebraic over $\mathbb{Q}$, and every element of $\mathbb{Q}(\sqrt2,\sqrt3)$ is built from them using field operations.
2. The extension $\mathbb{F}_{p^n}/\mathbb{F}_p$ is algebraic (indeed finite), since $[\mathbb{F}_{p^n}:\mathbb{F}_p]=n$.
3. If $\overline{\mathbb{Q}}$ denotes an algebraic closure of $\mathbb{Q}$, then $\overline{\mathbb{Q}}/\mathbb{Q}$ is algebraic by definition: its elements are precisely the complex numbers algebraic over $\mathbb{Q}$.