DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-26

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Kummer extensions from adjoining $n$ -th roots over a base field containing $μ_{n}$

Definition

Let $F$ be a field, let $n \geq 1$ , assume $char F ∤ n$ , and assume $F$ contains the full group $μ_{n}$ of $n$ -th roots of unity (The group $μ_{n} (K)$ of $n$ -th roots of unity in a field, and primitive $n$ -th roots of unity).

For a subgroup $B$ with

$(F^{\times})^{n} \subseteq B \subseteq F^{\times}$

and finite quotient $B / (F^{\times})^{n}$ , choose for each $b \in B$ an $n$ -th root $b^{1 / n}$ in an algebraic closure and write

$F (B^{1 / n}) : = F (b^{1 / n} : b \in B) .$

A finite Galois extension $K / F$ is a Kummer extension of exponent dividing $n$ when $K = F (B^{1 / n})$ for some such subgroup $B$ .

The cyclic one-generator case is the earlier theorem If $μ_{n} \subseteq F$ and $char F ∤ n$ , then a degree- $n$ extension is cyclic exactly when it is $F (α)$ with $α^{n} \in F$ and $x^{n} - α^{n}$ irreducible.

Depends on

Used by

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

B. Conrad, Kummer Theory, Theorem 5.12 (standard reference, not scraped)
J. Ash, Basic Abstract Algebra, Section 6.7 (standard reference, not scraped)

Kummer extensions from adjoining nn-th roots over a base field containing μnμn​