Jacques Charles Francois Sturm (1803-1855) finds the number of real roots of a given polynomial in a given interval.

Carl Gustav Jacobi (1804-1851) studies modular equations for elliptic functions which are fundamental for Hermite's 1858 solution of quintics.

Augistin-Louis Cauchy (1789-1857) determines how many roots of a polynomial lie inside a given contour in the complex plane.

Evariste Galois (1811-1832) writes down the main ideas of his theory in a letter to Auguste Chevalier the day before he dies in a duel.

Friedrich Julius Richelot (1808-1875) solves the cycolotomic equation:

in square roots.

George Birch Jerrard (1804-1863) shows that every quintic can be transformed to:

Karl Heinrich Graeffe (1799-1873) invents a widely used method to determine numerical roots by hand. Similar ideas had already been suggested independently by Edward Waring (1734-1798), Germinal Pierre Dandelin (1794-1847), Moritz Abraham Stern (1807-1894), and Nickolai Lobachevski (1792-1856). Johann Franz Encke (1791-1865) later perfects the method.