TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge 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.

Artin's fixed-field theorem: $[K : K^{G}] = ∣ G ∣$ and $Aut (K / K^{G}) = G$

Statement

If $G$ is a finite group of automorphisms of $K$ , then $[K : K^{G}] = ∣ G ∣$ and $Aut (K / K^{G}) = G$ .

Facts & Assumptions

Given: A field $K$ , a finite automorphism group $G$ , and the fixed field $K^{G}$ .

[L1]

If $G$ is a finite group of automorphisms of $K$ , then $[K : K^{G}] \geq ∣ G ∣$ (Artin's fixed-field lower bound $[K : K^{G}] \geq ∣ G ∣$ ).

[L2]

If $G$ is a finite group of automorphisms of $K$ , then $[K : K^{G}] \leq ∣ G ∣$ (Artin's fixed-field upper bound $[K : K^{G}] \leq ∣ G ∣$ ).

[L3]

For a finite extension $K / E$ , one has $∣ Aut (K / E) ∣ \leq [K : E]$ ( $Aut (K / F)$ is a group and $∣ Aut (K / F) ∣ \leq [K : F]_{s} \leq [K : F]$ ).

Proof

technique · direct

1.1L1L2

The opposing bounds [L1] and [L2] give $[K : K^{G}] = ∣ G ∣$ . In particular $K / K^{G}$ is finite. For $G = {1}$ this says $K^{G} = K$ and the degree is $1$ .

2.1step 1.1L3algebra∎

Every $σ \in G$ fixes $K^{G}$ , so $G \subseteq Aut (K / K^{G})$ . By [L3] and step 1.1, $∣ Aut (K / K^{G}) ∣ \leq [K : K^{G}] = ∣ G ∣$ ; a finite set containing $G$ and having at most its cardinality equals $G$ .

Artin's fixed-field theorem: $[K : K^{G}] = ∣ G ∣$ and $Aut (K / K^{G}) = G$

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Artin's fixed-field theorem: [K:KG]=∣G∣[K:KG]=∣G∣ and Aut⁡(K/KG)=GAut(K/KG)=G

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Artin's fixed-field theorem: $[K : K^{G}] = ∣ G ∣$ and $Aut (K / K^{G}) = G$