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.

The ring of integers has rank the degree

Statement

$O_{K}$ is a free $Z$ -module of rank $[K : Q]$ .

Facts & Assumptions

Given: A number field $K$ .

[F1]

Its integral closure is finite over $Z$ (Finite separable integral closures over normal Noetherian domains are module-finite).

[F2]

A submodule of finite free module over a PID is free (A submodule of a free module of finite rank over a PID is free of no larger rank).

[F3]

Every element of $K$ has a positive integer multiple in $O_{K}$ , so $K = Frac (O_{K})$ (Clearing denominators for an algebraic number).

Proof

technique · direct

1.1

Fact [F1] makes $O_{K}$ finitely generated; it is torsion-free because it is contained in the characteristic-zero field $K$ .

F1given

2.1

The PID structure theorem, equivalently [F2] after embedding in a finite free module, makes it free. Fact [F3] gives $O_{K} \otimes_{Z} Q ≅ K$ , so its rank is $[K : Q]$ .

F2F3step 1.1∎

Depends on

Used by

Discriminant of a basis and order Definition
Number-field discriminant is well-defined and nonzero Theorem
Orders have integral bases and finite index Theorem

Dependency tree · two levels

11 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, Proposition 2.29 (standard reference, not scraped)