## Find a polynomial over ZZ having a given set of roots

Find a polynomial in $\mathbb{Z}[x]$ satisfied by the roots of $q(x) = x^4 + \sqrt{3} x^2 + i$.

Note that the minimal polynomial of $\sqrt{3}$ over $\mathbb{Q}$ is $x^2 - 3$, and of $i$ is $x^2 + 1$. The conjugates of $\sqrt{3}$ are $\pm \sqrt{3}$, and of $i$ are $\pm i$. Note that any root of $q(x)$ is also a root of $p(x) = \prod_{a,b=0}^1 (x^4 + (-1)^a \sqrt{3} x^2 + (-1)^b i)$; evidently $p(x) = x^{16} - 6x^{12} + 11x^8 + 6x^4 + 1$. (WolframAlpha agrees.)

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