Next: The work of Vandermonde Up: The theory of permutations Previous: The degree of the

## The work of Lagrange

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.

Leo Liberti
Thu Feb 26 17:04:11 CET 1998