The general solution of the quadratic equation was found more than 4000 years ago. Mathematicians struggled for centuries to find formulas for the solutions of equations of higher degree, but despite the efforts of Euler , Bezout, Malfatti, Lagrange , and others, no general solutions were found. By 1832 Galois had developed the theory of Galois groups and described exactly when a polynomial equation is solvable.
Gray squares represent quintics that factor into lower degree polynomials.
The dark and light blue squares correspond to quintics that cannot be solved in radicals. There are polynomials of degree greater than four which do not factor over the rationals, but that can still be solved in radicals. In 1683 Tschirnhaus published Method of Eliminating All Intermediate Terms from a Given Equation . Although the title is exaggerated, Tschirnhaus's paper was the most important idea for the solution of algebraic equations in about 200 years. The b j can ultimately be expressed in radicals in terms of the a j .