DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

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The Lagrange resolvent attached to a cyclic action and a root of unity

Definition

Let $K / F$ be a cyclic extension of degree $n$ with generator $σ$ , and let $ζ$ be an $n$ -th root of unity in an overfield of $K$ (The group $μ_{n} (K)$ of $n$ -th roots of unity in a field, and primitive $n$ -th roots of unity).

For $x \in K$ , the Lagrange resolvent of $x$ attached to $(σ, ζ)$ is

$R_{σ, ζ} (x) : = \sum_{i = 0}^{n - 1} ζ^{- i} σ^{i} (x) .$

This is an $F (ζ)$ -linear expression in the orbit of $x$ under the cyclic action. When $ζ \in F$ , the resolvent lies in $K$ itself.

Depends on

Used by

Dependency tree · two levels

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Sources

J. S. Milne, Fields and Galois Theory, v5.10, Proposition 5.27 (standard reference, not scraped)
J. Ash, Basic Abstract Algebra, Section 6.7 (standard reference, not scraped)