Lemma

Let $\mathbb{F}$ be a field containing field $K$. Suppose $[\mathbb{F}:K]=n$ (i.e. the dimension of $\mathbb{F}$, the vector space is $n$). Let $\alpha\in\mathbb{F}$, $\alpha\not=0$. Suppose $m(x)$ is the minimal polynomial of $\alpha$. If $f(x)\in K[x]$ is s.t. $f(\alpha)=0$. Then $$m(x)|f(x)$$in $K[x]$.