Algebra theorem question

[Poly]

In theorem : Let F be a field and f(x) a nonconstant polynomial of degree n in F(x). Then there exists a splitting field K of f(x) over F s.t. [K:F]<=n! where [K:F] is dimension of K over F. My question : why is n! not n+1. Idea : Let K=F(u_1,u_2,...,u_n) where u_i is root of f(x) for all i. If the set {1_F,u_1,u_2,...,u_n} is linear independence then dim(K)=n+1, if not dim(K)<=n+1