LemmaStatement: 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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A nonzero number-field ideal has finite quotient

Statement

Every nonzero integral ideal of $O_{K}$ has finite additive quotient.

Proof

Given: $0 \neq a \subseteq O_{K}$ and an integral basis of rank $n$ .

1.1givenconstruct

Choose $0 \neq a \in a$ . Multiplication by $a$ is an injective endomorphism of the free rank- $n$ lattice $O_{K}$ , with nonzero integral determinant.

2.1step 1.1algebra∎

Its image $(a)$ has finite index, and $(a) \subseteq a \subseteq O_{K}$ ; thus $O_{K} / a$ is a quotient of the finite group $O_{K} / (a)$ .

Depends on

The ring of integers has rank the degree

Used by

Nontrivial number fields have discriminant of absolute value greater than one Corollary
The norm of a prime ideal Corollary
S-integers and S-units of a number field Definition
Class group of Q(sqrt 10) Example
Minkowski bound for Gaussian integers Example
Signature constant rules out discriminant ±1 Example
Finitely many ideals of bounded norm Lemma
Covolume of an integral ideal lattice Theorem
Ideal norm is multiplicative Theorem
Integral ideal factorisation in a number field, in ZF Theorem
Ramification is detected by the number-field discriminant Theorem
The norm of a principal integral ideal Theorem

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

J. S. Milne, Algebraic Number Theory, Chapter 4 (standard reference, not scraped)