Suppose that the minimal polynomial of $x$ in $\mathbb{Q}[X]$ is $$\sum_{n=0}^k a_n X^n.$$ How can I find the minimal polynomial of $ix$ (where $i$ is the imaginary unit) in $\mathbb{Q}[X]$ from the coefficients $a_n$?
I ran into a problem because some powers of $i$ (that is, $i$ and $-i$) are not in $\mathbb{Q}$.
Suppose you have $y=ix$ you can express that in rational coefficients as $x^2+y^2=0$. Now that's a polynomial in $x$ and your original minimal polynomial is another polynomial in $x$. You can compute the resultant of both these polynomials; it has to be zero for both polynomials to have a common zero, i.e. for there to be an $x$ which satisfies them both. The resultant is a determinant in the coefficients of the individual polynomials, but since one of these coefficients is $y^2$ that gives you a polynomial in $y$ which can be used to characterized $y$. Whether it's minimal or not is yet another question, so you best factor the result and see.