TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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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

1.1

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 $1$ .

L1L2

2.1

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 $G$ .

step 1.1L3algebra∎

Depends on

Used by

Distinct finite automorphism groups have distinct fixed fields Corollary
For a finite extension, |Aut(K/F)| divides [K:F] Corollary
Equivalent characterizations of a finite Galois extension Theorem
The fundamental theorem of finite Galois theory Theorem
The Galois translation theorem Theorem

Dependency tree · two levels

10 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, Theorem 3.4 and Corollary 3.5 (standard reference, not scraped)
K. Conrad, The Galois Correspondence, Theorems 5.2-5.3 (standard reference, not scraped)

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