Abel–Ruffini theorem (quintic not solvable by radicals)
Statement
There is no general solution in radicals—no finite formula using only coefficients of the polynomial together with the field operations (+, −, ×, ÷) and nested nth roots—that expresses the roots of an arbitrary quintic polynomial equation with rational (or general) coefficients. Equivalently, the Galois group of the general quintic over the function field of its coefficients is not solvable, so the equation cannot be reduced to a tower of quadratic/cubic extensions by adjunction of radicals alone. Paolo Ruffini published an early incomplete argument (1799); Abel gave the first widely accepted complete proof (1824); Galois later placed the phenomenon in Galois-theoretic language. The result is a theorem of pure algebra; formalizable in ZFC.
Formula
∄ universal radical tower over ℚ(a₀,…,a₄) expressing roots of x⁵ + a₄x⁴ + ··· + a₀ = 0 for generic coefficients
Proof sketch
Modern standard proof: identify the Galois group of the general quintic with S₅, which is not solvable; hence no radical tower can split the polynomial. Abel’s original memoir uses a direct algebraic attack on the hypothetical existence of a radical solution. See Tignol, Galois’ Theory of Algebraic Equations, or van der Waerden, Algebra.
Source
Abel, N. H. Mémoire sur les équations algébriques, où l’on démontre l’impossibilité de la résolution de l’équation générale du cinquième degré, 1824 (see Œuvres complètes). Ruffini, P. Teoria generale delle equazioni, 1799. Galois, É. Sur les conditions de résolubilité des équations par radicaux, 1832 (posthumous).
Domains
mathematics
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- Theorem: Abel–Ruffini theorem (quintic not solvable by radicals)
- Proof: Abel–Ruffini theorem (quintic not solvable by radicals)