Why there is no formula for the quintic
Show that there is no formula, built from the coefficients with addition, subtraction, multiplication, division and radicals, that gives the roots of every polynomial equation of degree five.
Posed by Joseph-Louis Lagrange, 1770
History and references
Solved in 1824 by Paolo Ruffini (1799, with a gap) and Niels Henrik Abel.
After Cardano's Ars Magna of 1545 gave formulas for degrees three and four, the quintic resisted for more than two centuries. Lagrange's study of 1770 explained why the known methods worked and why they stall at degree five. Ruffini published an impossibility proof in 1799 that had a gap, and Abel gave a complete one in 1824. Galois then described exactly which equations can be solved by radicals.
References
Solutions 0
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