Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck 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.

FALSE: every algebraic extension is simple

Statement

Every algebraic field extension is simple.

Facts & Assumptions

Given: The universal claim in the Statement and a prime p.

[L1]

The algebraic closure Fp‾/Fp is algebraic and infinite (Fp‾ is the union of its finite subfields and is an infinite algebraic extension).

[L2]

An algebraic element generates a finite simple extension (An element is algebraic over F if and only if its simple extension F(a)/F is finite).

Refutation

technique · direct
1.1L2

If Fp‾=Fp(α) for one element α, then α is algebraic and [L2] would make the extension finite.

2.1step 1.1L1∎

This contradicts the infinitude in [L1]. Hence the algebraic extension Fp‾/Fp is not simple, refuting the Statement.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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