CorollaryStatement: 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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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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For a finite extension, $∣ Aut (K / F) ∣$ divides $[K : F]$

Statement

If $K / F$ is a finite extension, then $∣ Aut (K / F) ∣$ divides $[K : F]$ .

Facts & Assumptions

Given: A finite extension $K / F$ , the finite group $G : = Aut (K / F)$ supplied by $Aut (K / F)$ is a group and $∣ Aut (K / F) ∣ \leq [K : F]_{s} \leq [K : F]$ , and its fixed field $E : = K^{G}$ (The fixed field $K^{G}$ of a group of field automorphisms).

[L1]

If $G$ is a finite group of automorphisms of $K$ , then $[K : K^{G}] = ∣ G ∣$ and $Aut (K / K^{G}) = G$ (Artin's fixed-field theorem: $[K : K^{G}] = ∣ G ∣$ and $Aut (K / K^{G}) = G$ ).

[L2]

If $F \subseteq E \subseteq K$ and both successive extensions are finite, then $[K : F] = [K : E] [E : F]$ (Tower law for finite extensions: $[L : F] = [L : K] [K : F]$ ).

Proof

technique · direct

1.1

Artin's theorem gives $∣ Aut (K / F) ∣ = ∣ G ∣ = [K : E]$ .

2.1

The tower formula gives $[K : F] = [K : E] [E : F] = ∣ Aut (K / F) ∣ [E : F]$ , proving the divisibility. Degree one and a trivial automorphism group both give divisor $1$ .

step 1.1L2∎

Depends on

Used by

FALSE: every degree-n extension has exactly n automorphisms False statement

Dependency tree · two levels

12 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

K. Conrad, The Galois Correspondence, Corollary 4.2 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, v5.10, Chapter 3 (standard reference, not scraped)

For a finite extension, ∣Aut⁡(K/F)∣∣Aut(K/F)∣ divides [K:F][K:F]