Gianfrancesco Malfatti (1731-1807), starting with a quintic, finds a sextic that factors if the quintic is solvable in radicals.

Lagrange finds a stationary solution of the three body problem that requires the solution of a quintic.

Erland Samuel Bring (1736-1798) proves that every quintic can be transformed to:

Jean Baptiste Joseph Fourier (1768-1830) determines the maximum number of roots in an interval.

Paolo Ruffini (1765-1822) publishes the book, General Theory of Equations, in which the Algebraic Solution of General Equations of a Degree Higher than the Fourth is Shown to Be Impossible.

Carl Friedrich Gauss (1777-1855) proves the fundamental theorem of algebra: Every nonconstant polynomial equation has at least one root.

Gauss solves the cyclotomic equation:

in square roots.

William George Horner (1768-1847) presents his rule for the efficient numerical evaluation of a polynomial. Ruffini had proposed a similar idea.

WilNiels Henrik Abel (1802-1829) publishes Proof of the Impossibility of Generally Solving Algebraic Equations of a Degree Higher than the Fourth.