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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The different in the monogenic case

Statement

If $O_{K} = Z [α]$ and $f$ is the monic minimal polynomial of $α$ , then $D_{K} = (f^{'} (α))$ .

Proof

Given: the power integral basis $1, α, \dots, α^{n - 1}$ .

1.1

Lagrange interpolation in the conjugates shows that the trace-dual lattice has $Z$ -basis $\frac{1}{f^{'} (α)}, \frac{α}{f^{'} (α)}, \dots, \frac{α^{n - 1}}{f^{'} (α)} .$

givenalgebra

2.1

Thus $O_{K}^{\lor} = f^{'} (α)^{- 1} O_{K}$ ; taking its fractional-ideal inverse yields $(f^{'} (α))$ .

step 1.1algebra∎

Depends on

Used by

A cyclotomic different preview Example
A quadratic-field codifferent Example

Dependency tree · two levels

7 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 4.3 (standard reference, not scraped)