It is not surprising that Galois was not understood even by the greatest mathematicians of the time. His papers make such a large number of unproved (though true) assumptions, that at first they really are incomprehensible. Modern Galois theory can be viewed from two possible angles, which are however closely linked: group theory and field theory. Following the group-theoretic approach, to each polynomial there corresponds a group of permutations of the roots; if the group contains a series of certain particular nested subgroups such that the intersection of the series is the identity element, this means that the identity is the only permutation of the roots that leave all the known rational relations between the roots valid, and hence it is possible to determine them. If the intersection contains other permutations apart from the identity, it means that all the known relations cannot determine the roots because even if we permute them with the permutations in the intersection the relations still hold. Taking the field-theoretic approach, for any polynomial having roots and coefficients in the field F there exists a field F' such that the polynomial is reducible into linear factors in F'. Obviously any field is such that the polynomial splits into linear factors. Any such field G is called a splitting field for f(x). It is clear that for any polynomial f(x) over F with roots the field is a splitting field for f. Very simply put, if F' is expressible by adjoining a finite number of radicals of the form to the base field F, then it is possible to express the roots in terms of the radical operations and hence there is a radical formula to find the roots in terms of the coefficients. Thus Galois found necessary and sufficient conditions for the roots to be expressed as rational functions of the coefficients, settling the problem definitively.