Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13
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.

The base field is the splitting field of the empty family and of every nonzero constant polynomial

Example

For every field F, the splitting field over F of the empty family is F. The same is true for every nonzero constant polynomial c∈F[x].

Facts & Assumptions

Given: A field F.

[F1]

A nonzero degree-zero polynomial splits as its leading coefficient times an empty product, and a splitting field is generated by the roots of the polynomial or family (Polynomials that split and splitting fields of a polynomial or a family of polynomials).

Verification

technique · direct
1.1

The empty family has no roots, and a nonzero constant also has no roots. In either case the subfield generated over F by the empty root set is F itself.

F1
2.1

The empty family is vacuously split over F, while c=c∏j=10(x−αj) is the required empty linear factorisation. Thus both cases satisfy the splitting-field definition.

F1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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