---
tag: 09EYMKMSB9
kind: definition
name: Algebraic extension
msc: [12F05]
defines: 0PTBM7K0R2
uses: [0WQ1A8SHTZ, 0NPW1DWECM]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_extension
    locator: lead section
---

## Definition

A [[field extension|0WQ1A8SHTZ]] $L/K$ is algebraic if every element of $L$ is [[algebraic|0NPW1DWECM]] over $K$.

## Remarks

[[Every finite extension is algebraic|0DFYTNZ6AM]], 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|02SJG1HC32]] 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.