I am in trouble to prove the following problem.
Let $F$ be a finite field and $K$ be an extension of $F$ such that $[K:F]=n$. Then show that for any monic irreducible polynomial $f\in F[x]$ of degree $d$ dividing $n$, there exists $\alpha\in K$ such that the minimal polynomial of $\alpha$ over $F$, $m_{F}(\alpha)$, is equal to $f$.
What I have been trying is as following. Firstly, I observed that $F$ should be a vector space over $\mathbb{F}_{p}$ for some prime $p$. Noting that $F$ is isomorphic to $\mathbb{F}_{p^m}$ for some positive integer $m$, I also noted that $K\cong \mathbb{F}_{p^{mn}}$. We know $\mathbb{F_{p^{nm}}}$ is Galois over $\mathbb{F_{p^m}}$ so I conclude that $K$ is Galois over $F$. And then I noted that $f$ should be a divisor of $x^{p^{nm}}-x$. But actually, I am not quite sure about it. I hope to have any help on this problem.