>>13804519Some exercises to get you started. Show that the coefficients of a unit polynomial are symmetrical polynomials in its roots. Conversely, show that every symmetrical polynomial in its roots is a polynomial in its coefficients. Now consider what happens when you use a proper sub-group of the full group of root symmetries. You should get larger sets of polynomials because they have to satisfy fewer conditions. Ultimately, even the roots themselves are 'symmetric' with respect to the identity subgroup of the symmetry group.
Galois theory says that under some conditions, every 'closed' set of polynomials in the roots comes from some subgroup of the symmetry group and vice versa. Also it notes that solving the equation by radicals is equivalent to finding a tower of subgroups of the symmetry group starting at the identity group and ending at the whole group, where each step has a normal subgroup and an abelian quotient. Since there is not such tower for S5, the general quintic has no solution by radicals.
To show that there are actually specific quintics with such a solution, you need to do some extra unpleasant grovelling about.