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 $q\in\mathbb{Z}$) ? ($i\in\mathbb{C}$) ?

Hint $\ f(q\,i) = a_0\! -\! a_2 q^2\! +\! a_4 q^4\! +\cdots + i\,q\,(a_1\! -\! a_3 q^2\! +\! a_5 q^4\! +\! \cdots) = g(q) + i\,q\,h(q)$
