Construct a degree 4 polynomial in having as a root. Construct a degree 6 polynomial having as a root.
For no particular reason whatsoever, let , , , and . Letting denote the elementary symmetric polynomials in four variables, and making these substitutions, we see that , , , and . Thus has as a root.
Note that is a root of and is a root of , and that the other roots of these polynomials are , , and . By Corollary 3.12 in TAN, is a polynomial in . A rather tedious calculation reveals that this polynomial is .