>>13720349how to prove quintic P(x) is insoluble in radical
construct the set Gal(K/Q) which is the automorphism group of splitting field Kover Q of polynomial P(x)
construct another set S which is the S5 permutation group of all the "good" permutations of (1,2,3,4,5)
after constructing these two sets, it's possible to prove that these two set are isomorphic as group
hence a contradiction can be derived as follow:
if P(x) is soluble in radical, then Gal(K/Q) is solvable group
but S5 is not solvable
so quintic is insoluble in radical
