# Solving the quintic by radicals

- kind: problem
- status: refuted (evidence tier V3)
- year: 1550
- authors: after Cardano's cubic and Ferrari's quartic formulas

**Statement.** Is there a formula in radicals for the roots of every polynomial of degree five?

There is not. Galois explained why: solvability corresponds to a property of the equation's symmetry group, and the symmetric group on five letters lacks it. Galois theory, the tool built to close this question, became one of the foundations of modern algebra.

*Evidence tier V3 is the import default for a result the curation records as settled in the published record. No formal proof or certificate is attached to this page yet.*

Posed 16th century, after Cardano's cubic and Ferrari's quartic formulas. Closed 1824, Niels Henrik Abel (after Ruffini, 1799); explained by Galois, 1832.

## Related problems

- The original curation links this problem to [Inverse Galois problem](/wiki/Problem:inverse-galois.md); the argument for the link has not been written yet.

## Sources

- [Wikipedia: Abel–Ruffini theorem](https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem)

[Algebra and groups](/wiki/Space:algebra.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations

- related to [inverse-galois](/wiki/Problem:inverse-galois.md) — The original curation links this problem to Inverse Galois problem; the argument for the link has not been written yet.

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Revision 51, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Imported from content/01-mathematics.json, curated for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (September 2026).. History: /wiki/Problem:quintic-insolvability?action=history. Edit: POST /api/wiki/page/Problem:quintic-insolvability {source, summary, base_rev: 51}.