Norm of $(\alpha - a) = (-1)^{\deg f}f(a)$

Question

Let $\Bbb Q(\alpha)$ be a number field, and $f$ the minimal polynomial of $\alpha$. Why is $N_{\Bbb Q(\alpha)/\Bbb Q} (\alpha-a)= (-1)^{\deg f}f(a)$?

This works obviously for $a=0$ by the definition of the norm. I know I saw a sleek argument once, but I can't seem to remember.

Answer 1

Hint. If $f(x)$ has roots $\alpha_1,\ldots,\alpha_n$, what is $$(a-\alpha_1)\cdots(a-\alpha_n)\ ?$$