Polynomials of degree ≤ 4 have closed-form radical solutions (quadratic formula, Cardano, Ferrari). The quintic x⁵+px+q generally does not — its Galois group is S₅, which is not solvable (has no chain of normal subgroups with abelian quotients). This visualizes the roots of a quintic family in the complex plane, shows the Galois group structure, and walks through why solvability fails.
Degree 3: Gal ⊆ A₃ or S₃; both solvable ✓ (Cardano)
Degree 4: Gal ⊆ S₄; has solvable series ✓ (Ferrari)
Degree 5: Generic Gal = S₅; S₅ is NOT solvable ✗
Why S₅ is not solvable:
Solvability requires a subnormal series with abelian quotients.
S₅ ⊃ A₅ ⊃ {e} — but A₅ is simple (no normal subgroups)!
A₅ is the smallest non-abelian simple group (order 60).
∴ no radical tower can express the roots in general.
Key insight: Permuting roots by radicals can only generate solvable subgroups. But Gal(generic quintic) = S₅ requires A₅ which resists this.