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

1.1F1given

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

2.1step 1.1algebra∎

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

Depends on

Used by

Discriminant of a quadratic field Corollary
Frobenius cycle type needs good reduction Counterexample
Mixing scaled and unscaled Minkowski covolumes fails Counterexample
Units of Z[√5] are a proper subgroup of the units of its maximal order Counterexample
Class group of Q(sqrt -5) Example
Class group of Q(sqrt 10) Example
Gaussian and eisenstein frobenius Example
Gaussian and Eisenstein integral bases Example
Minkowski bound for Gaussian integers Example
Real quadratic units and Pell's equation Example
The integral basis and discriminant of Q(sqrt 5) Example
Units of ℚ and the imaginary quadratic fields 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)