If p(x) is a polynomial of degree n in F[x] and if E is the splitting field of p(x) over F, then (E:F) is a divisor of n!
If the polynomial p(x) is separable or characteristic of F is 0, then it can be shown directly like this.
If E is the splitting field of p(x) in F[x], then E is the Galois extension of F. Thus |Gal(E/F)| = [E:F]. Since Gal(E/F) is the subgroup of S_n (link), [E:F] should be a divisor of n! by Lagrange's theorem.
prove it by induction over it's obvious for for consider two cases:
case 1: is irreducible. in this case for some and we have because is irreducible. also note that is a splitting
field for and thus, by induction hypothesis, divides therefore
case 2: is reducible. then for some with and we also have let be the splitting field for
which is contained in then would be a splitting field for thus, by induction hypothesis: