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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