False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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 finite extension satisfies $[K : F]_{s} = [K : F]$

Statement

Every finite extension $K / F$ satisfies $[K : F]_{s} = [K : F]$ .

Facts & Assumptions

Given: The universal equality in the Statement.

[L1]

The finite extension $F_{p} (t) / F_{p} (t^{p})$ has ordinary degree $p$ and separable degree $1$ ( $F_{p} (t) / F_{p} (t^{p})$ is purely inseparable of degree $p$ and separable degree one).

Refutation

technique · direct

1.1L1algebra

Since every prime $p$ is greater than $1$ , the two degrees in [L1] are unequal.

2.1step 1.1∎

This finite extension refutes the universal equality.

Depends on

$\mathbb F_p(t)/\mathbb F_p(t^p)$ is purely inseparable of degree $p$ and separable degree one

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

P. L. Clark, Field Theory, Chapter 4 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Chapters 3 and 5 (standard reference, not scraped)

FALSE: every finite extension satisfies [K:F]s=[K:F][K:F]s​=[K:F]