TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedaudited 2026-09-07

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.

Ideal norm is multiplicative

Statement

For nonzero integral ideals $a, b$ of $O_{K}$ , $N (a b) = N a N b$ .

Proof

Given: nonzero integral ideals $a, b$ .

1.1

Factor both ideals over their finite union of prime supports. The DVR calculation in the factorisation proof makes $p^{r} / p^{r + 1}$ one-dimensional over $O_{K} / p$ , so every successive quotient has cardinality $N p$ .

givenalgebra

2.1

Hence $N (p^{r + s}) = N (p^{r}) N (p^{s})$ at every prime; multiply over the finite support to obtain the claim.

step 1.1algebra∎

Depends on

Used by

Discriminant valuations from different exponents Corollary

Dependency tree · two levels

5 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 4.2(a) (standard reference, not scraped)