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.

The prime support of the different is ramification

Statement

A nonzero prime $P$ of $O_{K}$ lying over the rational prime $p$ divides $D_{K}$ if and only if $e (P / p) > 1$ , that is, if and only if $P$ itself ramifies.

Facts & Assumptions

Given: $P ∣ p$ and $e = e (P / p)$ .

[L1]

Theorem 4.13 in the cited source says that the exponent of $P$ in the different is $e - 1$ when $p ∤ e$ , and is at least $e$ when $p ∣ e$ .

Proof

1.1

Put $d = v_{P} (D_{K})$ . The prime $P$ divides the different exactly when $d > 0$ .

givenalgebra

2.1

If $e = 1$ , then $p ∤ e$ and [L1] gives $d = e - 1 = 0$ . If $e > 1$ , then [L1] gives either $d = e - 1 > 0$ or $d \geq e > 0$ , according as $p ∤ e$ or $p ∣ e$ . This proves the stated local equivalence.

L1step 1.1cases∎

Depends on

Used by

Different exponents in tame and wild ramification Theorem

Dependency tree · two levels

13 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

Keith Conrad, The Different Ideal, Theorem 4.13 (standard reference, not scraped)