Finite extension

Definition

A field extension $L/K$ is finite if its degree $[L : K]$ is finite. Otherwise $L/K$ is an infinite extension.

Remarks

A finite extension is not the same as an extension of finite fields. For example, $\mathbb{C}/\mathbb{R}$ is finite, and both fields are infinite. The extension $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is finite of degree $3$. The field $K(x)$ of rational functions is an infinite extension of $K$, because $1, x, x^2, \dots$ are linearly independent over $K$. Finite extensions are algebraic.

Explanations

L1 human

A finite extension is one that is a finite-dimensional vector space over the smaller field. Each element of it is a combination of finitely many fixed elements, with coefficients in the smaller field.