---
tag: 02NBKVQFRQ
kind: definition
name: Intermediate field
msc: [12F05]
defines: 0Z409N8RDG
uses: [0WQ1A8SHTZ, 07QY5RPFBR]
sources:
  - url: https://en.wikipedia.org/wiki/Field_extension
    locator: Extension field
---

## Definition

Let $L/K$ be a [[field extension|0WQ1A8SHTZ]].
An intermediate field of $L/K$ is a [[subfield|07QY5RPFBR]] $E$ of $L$ with $K \subseteq E$.

## Remarks

Then $L/E$ and $E/K$ are field extensions, and one calls $K \subseteq E \subseteq L$ a tower of fields.
The fields $K$ and $L$ are intermediate fields of $L/K$.
For example, $\mathbb{Q}(\sqrt{2})$ is an intermediate field of $\mathbb{R}/\mathbb{Q}$.
The [[fundamental theorem of Galois theory|0T1Q8D0FW8]] describes the intermediate fields of a Galois extension.

## Explanations

### L1 human

An intermediate field sits between the small field and the big field.
Galois theory gives a way to list all such fields, by a correspondence with subgroups of a group.