TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The norm of a principal integral ideal

Statement

For $0 \neq α \in O_{K}$ , $N ((α)) = ∣ N_{K / Q} (α) ∣$ .

Proof

Given: $0 \neq α \in O_{K}$ and an integral basis.

1.1givenalgebra

Multiplication by $α$ is an injective integer matrix on the integral lattice, and its image is $(α)$ .

2.1step 1.1algebra∎

The index of the image of an injective integer matrix is the absolute determinant; that determinant is the field norm. Hence the quotient cardinality has the asserted value.

Depends on

Used by

Class group of Q(sqrt -5) Example
Class group of Q(sqrt 10) Example
Minkowski bound for ideal classes Theorem
Product formula for a number field Theorem
The logarithmic unit image is a full lattice Theorem

Dependency tree · two levels

7 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

J. S. Milne, Algebraic Number Theory, Proposition 4.1(c) (standard reference, not scraped)