ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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$x^{5} - 2$ over $Q$ is solvable by radicals although it is a quintic

Example

The quintic polynomial $x^{5} - 2$ over $Q$ is solvable by radicals. Its splitting field is contained in

$Q (ζ_{5}, \sqrt[5]{2}),$

so it lies in a radical extension of $Q$ .

Facts & Assumptions

Given: The polynomial $f (x) = x^{5} - 2$ .

[F1]

A polynomial is solvable by radicals when its splitting field lies in a radical extension (A polynomial is solvable by radicals when its splitting field lies in a radical extension).

[L1]

A splitting field is generated by all roots of the polynomial (Polynomials that split and splitting fields of a polynomial or a family of polynomials).

Verification

technique · direct

1.1

The five roots of $x^{5} - 2$ are $\sqrt[5]{2}, ζ_{5} \sqrt[5]{2}, ζ_{5}^{2} \sqrt[5]{2}, ζ_{5}^{3} \sqrt[5]{2}, ζ_{5}^{4} \sqrt[5]{2} .$ Therefore [L1] makes the splitting field a subfield of $Q (ζ_{5}, \sqrt[5]{2})$ .

2.1

The field $Q (ζ_{5}, \sqrt[5]{2})$ is radical over $Q$ : first adjoin $ζ_{5}$ , a root of $x^{5} - 1$ , and then adjoin $\sqrt[5]{2}$ , a root of $x^{5} - 2$ . Hence [F1] says that $x^{5} - 2$ is solvable by radicals.

F1step 1.1∎

Depends on

Used by

FALSE: a polynomial solvable by radicals must have abelian Galois group False statement
FALSE: every quintic is insoluble by radicals False statement

Dependency tree · two levels

6 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, Fields and Galois Theory, v5.10, Section 7 (standard reference, not scraped)
J. Ash, Basic Abstract Algebra, Section 6.8 (standard reference, not scraped)

x5−2x5−2 over QQ is solvable by radicals although it is a quintic