TheoremStatement: 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.

Ramification is detected by the number-field discriminant

Statement

A rational prime $p$ ramifies in $K / Q$ if and only if $p ∣ d_{K}$ .

Proof

Given: an integral basis of $O_{K}$ .

1.1

Reducing its trace-pairing matrix modulo $p$ , a nontrivial radical is equivalent to failure of the residue algebra to be a product of separable fields, hence to some ramification index exceeding $1$ .

givenalgebra

2.1

The determinant of that matrix is $d_{K}$ , so the radical is nontrivial exactly when $p ∣ d_{K}$ . This proves both directions.

step 1.1algebra∎

Depends on

Used by

Only finitely many primes ramify Corollary

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, Theorem 3.35 (standard reference, not scraped)