DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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The norm on the explicit order $Z [\sqrt{D}]$

Definition

Fix a positive integer $D$ that is not a square. Write $Z [\sqrt{D}] : = {x + y \sqrt{D} : x, y \in Z},$ viewed only as the explicit rank-two order of such expressions. Its coordinate operations are

(x + y \sqrt{D}) + (u + v \sqrt{D}) = (x + u) + (y + v) \sqrt{D}

and

(x + y \sqrt{D}) (u + v \sqrt{D}) = (x u + D y v) + (x v + y u) \sqrt{D} .

The conjugate of $α = x + y \sqrt{D} \in Z [\sqrt{D}]$ is $\overline{α} : = x - y \sqrt{D} .$ The Pell norm of $α$ is $N_{D} (α) : = α \overline{α} = x^{2} - D y^{2} .$

Because $D$ is not a square, $\sqrt{D}$ is irrational, so equal expressions $x + y \sqrt{D} = u + v \sqrt{D}$ have equal coefficients $x = u$ and $y = v$ .

Used by

The elementary Pell order can miss units from the larger quadratic order Counterexample
The Pell norm is multiplicative Lemma
Integral Pell solutions form an abelian group Proposition
Generalized Pell solutions fall into finitely many Pell orbits Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Peter Hackman, Elementary Number Theory (standard reference, not scraped)
Keith Conrad, Pell's Equation, I (standard reference, not scraped)

The norm on the explicit order Z[D]Z[D​]

Definition

Used by

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Sources

The norm on the explicit order $Z [\sqrt{D}]$