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.

Integral and power integral bases

Definition

Let $O$ be an order in a number field $K$ of degree $n$ . An integral basis of $O$ is an ordered $Z$ -basis of $O$ . A power integral basis of $O$ is an integral basis of the form $(1, α, \dots, α^{n - 1})$ ; equivalently, $O = Z [α]$ . In the special case $O = O_{K}$ , this says $O_{K} = Z [α]$ , and then $K$ is called monogenic. A power integral basis is extra structure, not an assumption on an arbitrary order or number field.

Depends on

Order in a number field

Used by

Discriminant of a basis and order Definition
Orders have integral bases and finite index Theorem

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

Milne, Bases section (standard reference, not scraped)