Showing that an element is in the ring of integers Given a number field $K = \mathbb{Q}(\alpha)$ for $\alpha$ a root of $x^3 + x^2 -2x + 8$, show that $\beta = \frac{4}{\alpha} \in \mathscr{O}_K$. 
I'm not looking for a solution to the problem, but rather some help getting started. I know that two methods could be to either write $\beta$ as a linear combination of $\{1, \alpha, \alpha^2\}$ or to find it's minimal polynomial and show that it is in $\mathbb{Z}[x]$, but I am having difficulty doing either of these things in this case. Am I missing something obvious? Is there a better method?
Thank you.

Finding some polynomial for $\beta$ is not that difficult:
$$\beta^3=\frac{64}{\alpha^3},\quad\beta^2=\frac{16\alpha}{\alpha^3},\quad \beta =\frac{4\alpha^2}{\alpha^3},\quad\beta^0=\frac{\alpha^3}{\alpha^3} $$
hence
$$ \beta^3-\beta^2+2\beta+8=\frac{64-16\alpha+8\alpha^2+8\alpha^3}{\alpha^3}=0$$
