False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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FALSE: every quintic is insoluble by radicals

Statement

False claim: every quintic is insoluble by radicals.

Facts & Assumptions

Given: The quintic polynomial $x^{5} - 2$ over $Q$ .

[L1]

The polynomial $x^{5} - 2$ is solvable by radicals ( $x^{5} - 2$ over $Q$ is solvable by radicals although it is a quintic).

Refutation

technique · direct

1.1

The polynomial $x^{5} - 2$ has degree five, so it is a quintic.

L1algebra

2.1

By [L1] it is solvable by radicals, so the statement that every quintic is insoluble by radicals is false.

step 1.1L1∎

Depends on

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

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)