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.