Integer polynomial with two aligned roots

Question

Anyone have an example of a monic polynomial $f(x) \in \mathbb{Z}[x]$ with $f(0)=1$ such that for $\alpha \in \mathbb{C}\backslash\mathbb{R}$ and $t \in R, t>0, t\neq 1$ both $\alpha$ and $t\alpha$ are roots of f(x)?

Answer 1

Hint: There are a minimum of four roots. What are they? Have you tried writing down the multiplication of the four terms and see where it leads? It looks to me like any $\alpha$ with integral (real part and modulus squared) and any integral $t$ (other than $1$) work.