Suppose `w` is a primitive n-th root of unity, let `r = Ord_n(p)`, `f(w)=w^p`(the frobenius map), how to prove that w,f(w),f^2(w),...f^(r-1)(w) are all different in `Fp(w)` so that the minimal monic polynomial of `w` over Fp factors into those r factors in `Fp(w)`?

Here if `w` is replaced with a primitive element of `Fp(w)` then it's easy to prove. But `w` is not a primitive element of `Fp(w)`, it's a primitive element of `Fp(w)/Fp`.