Fixed field

Definition

Let $L$ be a field 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 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, 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, fixed fields turn subgroups into intermediate fields.