Let $i$ be a root of the equation $x^{2} + 1 = 0$ and let $\omega$ be a root of the equation $x^{2} + x + 1 = 0$. Construct a polynomial
$$f(x) = a_{0} + a_{1}x + \ldots + a_{n}x^{n}$$
where $a_{0}, a_{1}, \ldots, a_{n}$ are all integers such that $f(i + \omega) = 0$.