Let be . Let be its Galois group, which acts faithfully on the set of complex roots of . Numbering the roots lets one identify with a subgroup of the symmetric group . Since factors as in , the group contains a permutation that is a product of disjoint cycles of lengths 2 and 3 (in general, when a monic integer polynomial reduces modulo a prime to a product of distinct monic irreducible polynomials, the degrees of the factors give the lengths of the disjoint cycles in some permutation belonging to the Galois group); then also contains , which is a transposition. Since is irreducible in , the same principle shows that contains a 5-cycle. Because 5 is prime, any transposition and 5-cycle in generate the whole group; see . Thus . Since the group is not solvable, the equation is not solvable in radicals.
let be  let be its galois group which acts faithfully on the set of complex roots of  numbering the roots lets one identify with a subgroup of the symmetric group  since factors as in  the group contains a permutation that is a product of disjoint cycles of lengths 2 and 3 (in general when a monic integer polynomial reduces modulo a prime to a product of distinct monic irreducible polynomials the degrees of the factors give the lengths of the disjoint cycles in some permutation belonging to the galois group); then also contains  which is a transposition since is irreducible in  the same principle shows that contains a 5-cycle because 5 is prime any transposition and 5-cycle in generate the whole group; see  thus  since the group is not solvable the equation is not solvable in radicals