---
tag: 0MCXX29HWY
kind: definition
name: Algebraic closure
msc: [12F05]
defines: 0X6AB5P3G1
uses: [09EYMKMSB9, 02SJG1HC32]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_closure
    locator: lead section
---

## Definition

Let $K$ be a [[field|0QA8CHH8MG]].
An algebraic closure of $K$ is a [[field extension|0WQ1A8SHTZ]] $\overline{K}/K$ such that $\overline{K}/K$ is [[algebraic|09EYMKMSB9]] and $\overline{K}$ is [[algebraically closed|02SJG1HC32]].

## Remarks

The field $\mathbb{C}$ is an algebraic closure of $\mathbb{R}$, by the fundamental theorem of algebra.
The field $\overline{\mathbb{Q}}$ of [[algebraic numbers|0QFTETCHKB]] is an algebraic closure of $\mathbb{Q}$. The field $\mathbb{C}$ is not, because $\mathbb{C}/\mathbb{Q}$ is not algebraic.
If $K$ is infinite, then $\overline{K}$ has the same cardinality as $K$. If $K$ is finite, then $\overline{K}$ is countably infinite.
An algebraic closure is a largest algebraic extension of $K$, and a smallest algebraically closed field that contains $K$.

## Explanations

### L1 human

The algebraic closure adds to a field all the roots of all its polynomials, and nothing else.
For the rationals, it gives the algebraic numbers, not all the complex numbers.