I need to prove this part of a theorem: given a field $K$ such that $|K| = p^n$, a subfield $H \subset K$, and $\xi$ a primitive element of $K$; i need to say that $H(\xi) \subseteq K$.
Of course $K$ contains all the polynomial in $H[ \xi ]$.
$\xi$ is a root of the polynomial $x^{|K| - 1} - 1 \in H[x]$, so $\xi$ is a root of a monic factor of $x^{|K| - 1} - 1 \in H[x]$ irreducible in $H[x]$.
Now, i think that i must require also that this monic factor has degree $n$. Is it right? If so, how to prove it?