DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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The fixed field $K^{G}$ of a group of field automorphisms

Definition

Let $K$ be a field and let $G$ be a subgroup of its automorphism group. The fixed field of $G$ is

$K^{G} : = {x \in K : σ (x) = x for every σ \in G} .$

This is a subfield of $K$ . Every field automorphism fixes $0$ and $1$ , so these elements lie in $K^{G}$ . If $x, y \in K^{G}$ and $σ \in G$ , then $σ (x - y) = x - y$ and $σ (x y) = x y$ ; if also $x \neq 0$ , then $σ (x^{- 1}) = σ (x)^{- 1} = x^{- 1}$ . Thus $K^{G}$ is closed under subtraction, multiplication, and inverses of nonzero elements. In particular, when $G \leq Aut (K / F)$ (Relative field automorphisms and $Aut (K / F)$ ), one has $F \subseteq K^{G} \subseteq K$ .

Depends on

Relative field automorphisms and $\operatorname{Aut}(K/F)$

Used by

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Fields and Galois Theory, v5.10, Section 3 (standard reference, not scraped)
K. Conrad, The Galois Correspondence, Sections 4-5 (standard reference, not scraped)

The fixed field KGKG of a group of field automorphisms