TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

Tower law for finite extensions: $[L : F] = [L : K] [K : F]$

Let $F \subseteq K \subseteq L$ be fields. If $K / F$ and $L / K$ are finite, then $L / F$ is finite and

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

Given: Finite extensions $K / F$ and $L / K$ .

[L1]

Products of an $F$ -basis of $K$ and a $K$ -basis of $L$ form an $F$ -basis of $L$ (Products of bases form a basis in a tower of finite extensions).

[L2]

technique · direct

1.1givenL2choose

Choose bases of sizes $m = [K : F]$ and $n = [L : K]$ .

2.1step 1.1L1

By [L1], their $m n$ pairwise products form an $F$ -basis of $L$ .

3.1step 2.1L2algebra∎

Hence $L / F$ is finite and [L2] gives $[L : F] = m n = [L : K] [K : F]$ .

Depends on