TheoremStatement: Literature-sourcedProof: Literature-sourcedjudge 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.

Integers in a quadratic field

Statement

For squarefree $d \neq 1$ , $O_{Q (\sqrt{d})} = Z [(1 + \sqrt{d}) / 2]$ if $d \equiv 1 (m o d 4)$ , and $Z [\sqrt{d}]$ otherwise.

Facts & Assumptions

Given: Squarefree $d \neq 1$ .

[F1]

Integral elements have integral trace and norm (Trace and norm of an algebraic integer).

Proof

technique · direct

1.1

Write an integral element as $(a + b \sqrt{d}) / c$ in lowest terms. Fact [F1] forces $2 a / c, (a^{2} - d b^{2}) / c^{2} \in Z$ .

F1given

2.1

These divisibilities force $c ∣ 2$ ; the $c = 2$ case occurs exactly when $d \equiv 1 (m o d 4)$ , yielding the displayed bases.

step 1.1algebra∎

Depends on

Used by

Discriminant of a quadratic field Corollary
Gaussian and Eisenstein integral bases Example
The integral basis and discriminant of Q(sqrt 5) Example

Dependency tree · two levels

9 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 2.12 (standard reference, not scraped)