## Construct a polynomial over ZZ having a given root

Construct a degree 4 polynomial in $\mathbb{Z}[x]$ having $\sqrt{2}+i$ as a root. Construct a degree 6 polynomial having $\sqrt{2}\sqrt[3]{3}$ as a root.

For no particular reason whatsoever, let $x_1 = \sqrt{2}+i$, $x_2 = \sqrt{2} - i$, $x_3 = - \sqrt{2} +i$, and $x_4 = -\sqrt{2} - i$. Letting $\sigma_i$ denote the elementary symmetric polynomials in four variables, and making these substitutions, we see that $\sigma_1 = 0$, $\sigma_2 = -2$, $\sigma_3 = 0$, and $\sigma_4 = 9$. Thus $p(x) = x^4 - 2x^2 + 9$ has $\sqrt{2}+i$ as a root.

Note that $\alpha_1 = \sqrt{2}$ is a root of $a(x) = x^2-2$ and $\beta_1 = \sqrt[3]{3}$ is a root of $b(x) = x^3-3$, and that the other roots of these polynomials are $\alpha_2 = -\sqrt{2}$, $\beta_2 = \frac{-1}{2} + \frac{\sqrt{3}}{2}i$, and $\beta_3 = \frac{-1}{2} - \frac{\sqrt{3}}{2}i$. By Corollary 3.12 in TAN, $\prod_i \prod_j (x-\alpha_i\beta_j)$ is a polynomial in $\mathbb{Z}[x]$. A rather tedious calculation reveals that this polynomial is $x^6-72$.