CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 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.

The norm of a prime ideal

Statement

For a nonzero prime $P \subseteq O_{K}$ , there is a rational prime $p$ and an integer $f \geq 1$ with $P \cap Z = (p)$ and $N P = p^{f}$ .

Proof

Given: a nonzero prime ideal $P$ .

1.1givenalgebra

The finite domain $O_{K} / P$ is a finite field; its characteristic is a rational prime $p$ and its kernel on $Z$ is $(p)$ .

2.1step 1.1algebra∎

As a finite-dimensional vector space over $F_{p}$ , that field has $p^{f}$ elements for some $f \geq 1$ , which is exactly $N P$ .

Depends on

Used by

Class group generated by small prime ideals Corollary
Higher-degree class group by norm exclusions Example
Number field places classification Theorem

Dependency tree · two levels

6 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)