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

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
1.1

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

L2
2.1

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

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 45 results over 10 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