False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck 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.

FALSE: every degree- $n$ extension has exactly $n$ automorphisms

Statement

False claim. Every finite extension $K / F$ has $∣ Aut (K / F) ∣ = [K : F]$ .

Facts & Assumptions

Given: The valid upper bound of $Aut (K / F)$ is a group and $∣ Aut (K / F) ∣ \leq [K : F]_{s} \leq [K : F]$ and divisibility result of For a finite extension, $∣ Aut (K / F) ∣$ divides $[K : F]$ .

[L1]

$Q (\sqrt[3]{2}) / Q$ has degree three and trivial automorphism group ( $Q (\sqrt[3]{2}) / Q$ is separable and nonnormal with trivial automorphism group).

Refutation

technique · direct

1.1

The extension in [L1] has degree $3$ but automorphism-group order $1$ , so the two numbers are unequal.

2.1

The same witness satisfies the correct statements $1 \leq 3$ and $1 ∣ 3$ , showing that equality, not the bound or divisibility, is the failed assertion.

step 1.1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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 4.3 and examples (standard reference, not scraped)

FALSE: every degree-nn extension has exactly nn automorphisms