TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Orders have integral bases and finite index

Statement

Every order has an integral basis and finite additive index in $O_{K}$ .

Facts & Assumptions

Given: An order $O \subseteq O_{K}$ .

[F1]

$O_{K}$ is free of rank $n$ (The ring of integers has rank the degree).

[F2]

Full-rank PID submodules admit simultaneous bases (Simultaneous bases for a submodule of a finite free module over a PID).

Proof

technique · direct

1.1

The order has full rank by definition, so [F1] makes it a rank- $n$ submodule of a free rank- $n$ group.

F1given

2.1

Apply [F2]: the quotient is a finite direct sum of $Z / a_{i} Z$ , and the resulting basis of $O$ is integral.

F2step 1.1∎

Depends on

Used by

Order-index discriminant formula Corollary

Dependency tree · two levels

12 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

Milne, Bases section (standard reference, not scraped)