= Niels Henrik Abel
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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.<Mémoire sur les équations algébriques by Niels Henrik Abel>[Original title and principal impossibility statement]{ref}
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.
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