Let K be a field, K[x] an irreducible polynomial of degree n 5 and

Let K be a field, ∫ ϵ K[x] an irreducible polynomial of degree n ≥ 5 and F a splitting field of ∫ over K. Assume that Aut_KF ≅ S_n. Let u be a root of ∫ in F. Then

(a) K(u) is not Galois over K; [K(u) : KJ = n and Aut_KK(u) = I (and hence is solvable).

(b) Every normal closure over K that contains u also contains an isomorphic copy of F.

(c) There is no radical extension field E of K such that E ⊃ K(u) ⊃ K.