On page 614 of Dummit and Foote's Abstract Algebra, while deriving the resolvent cubic for a quartic $g$, they go from a trio of elements $\theta_i, i=1,2,3$ which were defined in terms of the roots of $g(y)= y^4 + py^2 + qy + r$ to an observation that since these are fixed under permutations of the roots, it follows that the elementary symmetric functions in the $\theta$'s are fixed by all permutations in $S_4$. So far, so good. But then they claim that "an elementary computation in symmetric functions shows that these elementary symmetric functions are $2p$, $p^2 - 4r$, and $-q^2$" and hence the $\theta$'s are roots of $x^3 - 2px^2 + (p^2-4r)x+q^2$.
This doesn't seem at all obvious to me. How did they go from a statement about the symmetric functions in the roots of a polynomial to deriving those functions in terms of the polynomial's coefficients?