Ludwig Sylow (1832-1918) puts the finishing touches on Galois's proofs on solvability.

Hermann Amandus Schwarz (1843-1921) investigates the relationship between hypergeometric differential equations and the group structure of the Platonic solids, an important part of Klein's solution to the quintic.

Felix Klein (1849-1925) solves the icosahedral equation in terms of hypergeometric functions. This allows him to give a closed-form solution of a principal quintic.

Ferdinand von Lindemann (1852-1939) expresses the roots of an arbitrary polynomial in terms of theta functions.

John Stuart Cadenhead Glashan (1844-1932), George Paxton Young (1819-1889), and Carl Runge (1856-1927), show that all irreducible solvable quintics with the quadratic, cubic, and quartic terms missing have a spezial form.

Vincenzo Mollame (1848-1912) and Ludwig Otto Hoelder (1859-1937) prove the impossibility of avoiding intermediate complex numbers in expressing the three roots of a cubic when they are all real.

Karl Weierstrass (1815-1897) presents an interation scheme that simultaneously determines all the roots of a polynomial.

David Hilbert (1862-1943) proves that for every n there exists an nth polynomial with rational coefficients whose Galois group is the symmetric group S_n