Niels Henrik Abel proved that the general quintic equation does not have solvability by radicals. His 1824 pamphlet established a limitation on formulas built from arithmetic operations and repeated extraction of roots, rather than a failure of numerical solution. Particular quintic equations can still have solvability by radicals; the impossibility concerns a formula for the general case.[ref]
Mémoire sur les équations algébriques by Niels Henrik Abel. Original title and principal impossibility statement
The result, expressed in the Abel-Ruffini theorem, helped shift the study of polynomial equations from searching for ever larger formulas toward asking which structural properties permit an algebraic solution.
Articles by others on the same topic
Niels Henrik Abel was a Norwegian mathematician who lived from August 5, 1802, to April 6, 1829. He is best known for his significant contributions to various areas of mathematics, particularly in the field of algebra. Abel's most notable accomplishment is his proof that the general quintic equation (the polynomial equation of degree five) cannot be solved in terms of radicals, which established an important result in the theory of equations.