How do you the simplified polynomial with integer coefficients and the following root(s):
I think there is something related to conjugates, but I do not know how to do this.
Due to the complex conjugates theorem,
$$((x-\sqrt3)-4i)((x-\sqrt3)+4i) = (x-\sqrt3)^2+16=x^2-2\sqrt3x+19$$ Here, there is a problem since there are no integer coefficients. I tried squaring the expression $x^2-2\sqrt3+19$ as I thought maybe $\sqrt3+4i$ and $\sqrt3-4i$ were double roots, but this was useless, since it produced more irrational solutions.
Does this mean that the expression with $\sqrt3+4i$ and $\sqrt3-4i$ as its roots does not have rational solutions? If not, how would I go about approaching this problem?