Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

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 Ks.

[L1]

The separable degree satisfies [K:F]s=[Ks:F] (For a finite extension, [K:F]s=[Ks:F]).

[L2]

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

Proof

technique · direct
1.1L2

The tower law [L2] gives [K:F]=[K:Ks][Ks:F].

2.1step 1.1L1algebra∎

Substituting [L1] yields [K:F]=[K:Ks][K:F]s, so [K:F]s divides [K:F].

Depends on

Used by

Dependency tree · two levels

9 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