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.

Discriminant of a basis and order

Definition

For an ordered $Q$ -basis $α_{1}, \dots, α_{n}$ , set $disc (α_{1}, \dots, α_{n}) = \det ({Tr}_{K / Q} (α_{i} α_{j}))_{i, j} .$ For an order $O \subseteq K$ , define $disc (O)$ to be the discriminant of any integral basis of $O$ . This is independent of the integral basis: the change matrix between two $Z$ -bases is unimodular, and Change of basis for discriminants ↗ multiplies the discriminant by the square of its determinant, which is $1$ . Finally, $d_{K} : = disc (O_{K})$ ; the existence of an integral basis of $O_{K}$ is supplied by The ring of integers has rank the degree.

Depends on

Used by

Discriminant of a quadratic field Corollary
Change of basis for discriminants Lemma
Embedding determinant formula Theorem

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, Discriminants section (standard reference, not scraped)