TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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.

The separable degree divides the degree of every finite extension

Statement

For every finite field extension $K / F$ , the natural number $[K : F]_{s}$ divides $[K : F]$ .

Facts & Assumptions

Given: A finite extension $K / F$ and its relative separable closure $K_{s}$ .

[L1]

The separable degree satisfies $[K : F]_{s} = [K_{s} : F]$ (For a finite extension, $[K : F]_{s} = [K_{s} : F]$ ).

[L2]

Ordinary degrees multiply in a finite tower (Tower law for finite extensions: $[L : F] = [L : K] [K : F]$ ).

Proof

technique · direct

1.1

The tower law [L2] gives $[K : F] = [K : K_{s}] [K_{s} : F]$ .

2.1

Substituting [L1] yields $[K : F] = [K : K_{s}] [K : F]_{s}$ , so $[K : F]_{s}$ divides $[K : F]$ .

step 1.1L1algebra∎

Depends on

Used by

The inseparable degree [K:F]ᵢ=[K:F]/[K:F]ₛ of a finite extension Definition

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 8 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, Chapters 4 and 5 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Chapters 3 and 5 (standard reference, not scraped)