I'm reading through Stewart's Galois Theory. In chapter one, the author asserts that for a cubic polynomial with roots $\alpha_0,\alpha_1,\alpha_2$, and $\omega$ a principal cube root of unity, consider
$u=(\alpha_0+ \omega \alpha_1 + \omega^2 \alpha_2)^3$
$v=(\alpha_0+ \omega \alpha_2 + \omega^2 \alpha_1)^3$
It can be shown that $u+v$ is a symmetric function of the roots, that is, the value is invariant under any permutation of the roots. Then the author asserts that the function can then be written as a rational function of the coefficients. In this particular example, I can work out the details and see why it works. But in general why is this assertion true for polynomials of any degree (or other symmetric functions) ?