Polynomial Has Root If and Only If It Is Divisible by Minimal Polynomial

Proof

Consider the result of dividing $p$ by $f_{\alpha }$, the minimal polynomial of $\alpha$:

Now, since $\mathrm{deg}(r)<\mathrm{deg}(f_{\alpha })$ we have that either $r=0$, which gives the divisibility we desire, or we have a contradiction, as:

which means since $p(\alpha )=f_{\alpha }(\alpha )=0$ that $r(\alpha )=0$, contradicting the minimality of $f_{\alpha }$.