Easy way to find roots of the form $qi$ of a polynomial Let $p$ be a polynomial over $\mathbb{Z}$, we know that there is an easy way to check if $p$ have rational roots (using the rational root theorem).
Is there an easy way to check if $p$ have any roots of the form $qi$ where $q\in\mathbb{Q}$ (or at least $q\in\mathbb{Z}$) ? ($i\in\mathbb{C}$) ?

You can actually do better than the normal Rational Root Theorem. Note that $p$ has integer (and thus real) coefficients. So if $ci/d$ is a root of $p$, then so is $-ci/d$. So $p$ must have $d^2x^2+c^2$ as a factor. Then you can conclude from the Gauss's lemma proof of the Rational Root Theorem that $d^2$ — not just $d$ — is a factor of the leading coefficient of $p$, and similarly $c^2$ is a factor of the trailing coefficient of $p$.
Combining this with Bill's/Karolis's answer will probably mean you have very few candidates to check even for moderately complicated polynomials.
