TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Different exponents in tame and wild ramification

Statement

For $P ∣ p$ , $v_{P} (D_{K}) \geq e (P / p) - 1$ . Equality holds when ramification is tame; when it is wild, $v_{P} (D_{K}) \geq e (P / p)$ .

Facts & Assumptions

[L1]

Lemma 4.12 and the proof of Theorem 4.13 in the cited source give the trace criterion for divisibility by powers of $P$ and apply it to the filtration of $O_{K} / P^{e}$ .

Proof

Given: a prime $P ∣ p$ and its ramification index $e$ .

1.1

That trace criterion first gives $v_{P} (D_{K}) \geq e - 1$ . For the next power, the successive quotients $P^{i} / P^{i + 1}$ are all isomorphic to the residue field as modules, so the relevant trace is $e {Tr}_{O_{K} / P / F_{p}} .$

L1givenalgebra

2.1

The trace of a finite separable field extension is not the zero map. Hence the display in step 1.1 vanishes identically exactly when $p ∣ e$ . Thus $p ∤ e$ gives exact exponent $e - 1$ , while $p ∣ e$ gives exponent at least $e$ , which are precisely the tame and wild cases.

L1step 1.1cases∎

Depends on

Used by

A tame different exponent Example
A wild different exponent Example

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

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