Joseph Louis Lagrange (1736-1813), a French mathematician, proved that a
polynomial equation can be solved by radicals if a particularly related
equation called the Lagrange resultant has degree less than the degree
of the original equation. The proof involved a concept that later was to
become fundamental to the theory of polynomial equations: that of the
permutation of the roots. The Lagrange resultant is a polynomial constructed
by means of a rational function of all possible permutations of the roots. An
important characteristic of the theorem is that it is an existence theorem, as
opposed to the previous methods that had all been concerned with the
construction of a solving formula. Using this result, Ruffini produced in 1799
an erroneous proof that a polynomial equation od degree greater than 4 was not
soluble by radicals. Abel independently found in 1824 a correct proof of the
same theorem; thus the question of solving polynomial equation was in part
settled. Still there are polynomials of degree greater than 4 that are clearly
soluble, like 
. What remained to be done was to find a method to
determine exactly which polynomial is soluble.