Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04
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.

A coefficient field maps isomorphically to the residue field

Statement

Let (A,m) be a local ring with residue field k=A/m. If K⊆A is a coefficient field, then the residue map restricts to a field isomorphism K→∼k.

Facts & Assumptions

Given: A local ring (A,m) and a coefficient field K⊆A.

[L1]

A coefficient field is defined to be a subfield on which the residue map is an isomorphism onto the residue field (Equicharacteristic local rings and coefficient fields).

Proof

technique · unpack the definition
1.1L1given

By [L1], the defining property of a coefficient field is precisely that the composite K↪A↠A/m is an isomorphism.

2.1step 1.1∎

Therefore a coefficient field maps isomorphically to the residue field.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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