CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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.

For every field $F$ , $F [x]$ is a principal ideal domain

For every field $F$ , every ideal of $F [x]$ is generated by one polynomial; equivalently, $F [x]$ is a principal ideal domain.

Given: A field $F$ .

[L1]

[L2]

Every Euclidean domain is a principal ideal domain (Every Euclidean domain is a principal ideal domain).

[L3]

A principal ideal domain is an integral domain in which every ideal is principal (Principal ideal domain).

technique · direct

1.1givenL1L2

By [L1] and [L2], $F [x]$ is a principal ideal domain.

2.1step 1.1L3∎

Unfolding [L3], every ideal of $F [x]$ therefore has the form $(d)$ for some polynomial $d$ .

Depends on