---
tag: 0NRYPM9Y27
kind: definition
name: Finite extension
msc: [12F05]
defines: 0C1Y04SZ1W
uses: [0GS74FGFEK]
sources:
  - url: https://en.wikipedia.org/wiki/Degree_of_a_field_extension
    locator: Definition and notation
---

## Definition

A [[field extension|0WQ1A8SHTZ]] $L/K$ is finite if its [[degree|0GS74FGFEK]] $[L : K]$ 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|0DFYTNZ6AM]].

## 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.