Algebraic field extension

Definition. A field extension $L/K$ is algebraic if every $\alpha\in L$ is algebraic over K . Equivalently, each $\alpha\in L$ satisfies some nonzero polynomial in $K[x]$.

Every finite extension (finite degree ) is algebraic, but an algebraic extension can be infinite degree.

See also. separable extension , normal extension , algebraic closure .

Examples.

1. $\mathbb{Q}(\sqrt2,\sqrt3)/\mathbb{Q}$ is algebraic (indeed finite).
2. $\overline{\mathbb{Q}}/\mathbb{Q}$ (the field of all algebraic numbers) is algebraic but has infinite degree.
3. $\mathbb{C}/\mathbb{R}$ is algebraic of degree $2$.