---
tag: 0KMJ6919WS
kind: definition
name: Fixed field
msc: [12F10]
defines: 0J1E5ZRZRM
uses: [07QY5RPFBR, 0B6QC0C0G1]
sources:
  - url: https://en.wikipedia.org/wiki/Fixed-point_subring
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Galois_extension
    locator: lead section
---

## Definition

Let $L$ be a [[field|0QA8CHH8MG]] and let $H$ be a group of ring automorphisms of $L$.
The fixed field of $H$ is
$$L^H = \{ a \in L : \sigma(a) = a \text{ for every } \sigma \in H \}.$$

## Remarks

The set $L^H$ is a [[subfield|07QY5RPFBR]] of $L$, because each $\sigma \in H$ respects the field operations.
For $L = \mathbb{C}$ and $H = \{\mathrm{id}, \sigma\}$, with $\sigma$ complex conjugation, $L^H = \mathbb{R}$.
If $H$ is finite, then $[L : L^H] = |H|$ and $L/L^H$ is a [[Galois extension|0DS2F7GK05]], by a theorem of Emil Artin.
More generally, a group acting on a ring has a fixed subring, the ring of invariants.

## Explanations

### L1 human

The fixed field is the part of the field that a group of symmetries does not move.
In the [[fundamental theorem of Galois theory|0T1Q8D0FW8]], fixed fields turn subgroups into intermediate fields.