DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Ring of integers

Definition

For a number field $K$ , its ring of integers is $O_{K}$ , the integral closure of $Z$ in $K$ . It is not an arbitrary order.

Depends on

Used by

Minimal-polynomial criterion for algebraic integers Corollary
Rings of integers are Dedekind domains Corollary
Order in a number field Definition
Integers of Q Example
Clearing denominators for an algebraic number Theorem
Integers in a quadratic field Theorem
The ring of integers has rank the degree Theorem

Dependency tree · two levels

6 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, Definition 2.5 (standard reference, not scraped)