Algebraic extension

Definition

A field extension $L/K$ is algebraic if every element of $L$ is algebraic over $K$.

Remarks

Every finite extension is algebraic, so $\mathbb{C}/\mathbb{R}$ and $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ are algebraic. The converse fails: the field of all algebraic numbers is an algebraic extension of $\mathbb{Q}$ of infinite degree. The extension $\mathbb{R}/\mathbb{Q}$ is not algebraic, because $\pi$ is transcendental. An algebraically closed field has no algebraic extension other than itself.

Explanations

L1 human

In an algebraic extension, every new element solves a polynomial equation over the base field. There are no elements that behave like free variables.