False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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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.

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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

1.1

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

L1algebra

2.1

This finite extension refutes the universal equality.

step 1.1∎

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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 29 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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]