CorollaryStatement: Literature-sourcedProof: AI-adaptedjudge 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.

Order-index discriminant formula

Statement

For an order $O \subseteq O_{K}$ , $disc (O) = [O_{K} : O]^{2} d_{K}$ .

Facts & Assumptions

Given: An order $O$ .

[F1]

Simultaneous bases exist with finite index (Orders have integral bases and finite index).

[F2]

Discriminants change by a determinant square (Change of basis for discriminants).

Proof

technique · direct

1.1

Choose the bases from [F1]; the absolute change determinant is the index.

F1given

2.1

Apply [F2] and square the determinant.

F2step 1.1∎

Depends on

Used by

Squarefree power discriminant criterion Corollary
Index 2 obstructs reading factorisation from Z[sqrt 5] modulo 2 Example
The integral basis and discriminant of Q(sqrt 5) Example
The nonmaximal quadratic order Z[sqrt 5] inside O_Q(sqrt 5) Example

Dependency tree · two levels

10 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, Remark 2.25 (standard reference, not scraped)