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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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 $Aut (K / F)$

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

Definition

Depends on

Dependency tree · 2 levels

Sources

The fixed field KGKG of a group of field automorphisms

Definition

Depends on

Dependency tree · 2 levels

Sources

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