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.

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