---
tag: 0WQ1A8SHTZ
kind: definition
name: Field extension
msc: [12F05]
defines: 0EQWVHF3W5
uses: [0QA8CHH8MG, 07QY5RPFBR]
sources:
  - url: https://en.wikipedia.org/wiki/Field_extension
    locator: Extension field
---

## Definition

Let $K$ and $L$ be [[fields|0QA8CHH8MG]].
The field $L$ is an extension of $K$ if $K$ is a [[subfield|07QY5RPFBR]] of $L$.
One writes $L/K$ for this field extension.

## Remarks

Often one also calls $L$ an extension of $K$ when there is an injective [[ring homomorphism|0GGE0Z0JQ1]] $K \to L$. Then one identifies $K$ with its image.
An extension $L/K$ makes $L$ a vector space over $K$. Its dimension is the [[degree of the extension|0GS74FGFEK]].
Examples: $\mathbb{C}/\mathbb{R}$, $\mathbb{R}/\mathbb{Q}$ and $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$.
The notation $L/K$ does not mean a quotient.

## Explanations

### L1 human

A field extension is a bigger field that contains a given field.
For example, the complex numbers extend the real numbers by an element $i$ with $i^2 = -1$.
Field theory studies how the bigger field is built from the smaller one.