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

Separable degree is multiplicative in finite towers: [L:F]s=[L:K]s[K:F]s

Statement

For every finite tower F⊆K⊆L,

[L:F]s=[L:K]s[K:F]s.

Facts & Assumptions

Given: A finite tower F⊆K⊆L and an algebraic closure Ω/F.

[L1]

Separable degree counts embeddings into an algebraic closure (The separable degree [K:F]s as a count of embeddings into an algebraic closure).

[L2]

Restriction from F-embeddings of L to F-embeddings of K is surjective, and every fibre has cardinality [L:K]s (Restriction partitions embeddings in a finite tower into extension fibres).

Proof

technique · direct
1.1L2

By [L2], the finite set Hom⁡F(L,Ω) is the disjoint union of the restriction fibres indexed by Hom⁡F(K,Ω).

2.1step 1.1L1L2algebra∎

There are [K:F]s fibres by [L1], and each has [L:K]s elements by [L2]. Counting the disjoint union gives the displayed product.

Depends on

Used by

Dependency tree · two levels

12 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