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

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An Eisenstein prime is totally ramified

Statement

If $L = K (α)$ is generated by a monic Eisenstein polynomial of degree $n$ at $p$ , then $p O_{L} = P^{n}$ for one prime $P$ ; thus it is totally ramified.

Proof

Given: the Eisenstein polynomial at $p$ .

1.1

Let $P ∣ p$ have ramification index $e$ . Comparing the valuations of the terms in the Eisenstein equation forces $n v_{P} (α) = e$ ; in particular $e \leq [L : K] \leq n$ .

givenalgebra

2.1

The inequalities in step 1.1 are therefore equalities, so $v_{P} (α) = 1$ and $e = n$ . The resulting factorisation is $p O_{L} = (p, α)^{n}$ ; hence this is the unique prime factor and the prime is totally ramified.

step 1.1algebra∎

Depends on

Dedekind--Kummer prime factorisation

Used by

An Eisenstein total-ramification calculation 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

J. S. Milne, Algebraic Number Theory, Proposition 3.53 (standard reference, not scraped)