Find a polynomial over ZZ having a given set of roots

Verify that the roots of q(x) = x^3 + (2-\sqrt{7})x + 1+3i are algebraic integers.


Note that the minimal polynomial of 2 - \sqrt{7} is a(x) = x^2 - 4x - 3 and of 1+3i is b(x) = x^2 - 2x + 10; so the conjugates of 2 - \sqrt{7} are 2 \pm \sqrt{7} and of 1 + 3i are 1 \pm 3i.

Every root of q(x) is also a root of \prod_{a,b=0}^1 (x^3 + (2 + (-1)^a \sqrt{7})x + 1 + (-1)^b3i); evidently, p(x) = x^{12}+8 x^{10}+4 x^9+10 x^8+24 x^7+20 x^5+105 x^4+16 x^3+208 x^2+80 x+100. (WolframAlpha agrees.)

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