Deriving Dummit's sextic resolvent for quintic polynomials

Question

I have been studying solutions of polynomials in radicals and am having difficulty in deriving Dummit's sextic resolvent for quintic polynomials as given in the paper,

Dummit, D. S. "Solving Solvable Quintics." Math. Comput. 57, 387-401, 1991.

http://www.ams.org/journals/mcom/1991-57-195/S0025-5718-1991-1079014-X/S0025-5718-1991-1079014-X.pdf

There he states,

"By computing the elementary symmetric functions of the $\theta_i $, which are symmetric polynomials in $x_1, x_2, x_3, x_4, x_5$, it is a relatively straightforward matter to express these elements in terms of $s_1, s_2, s_3, s_4, s_5$ to determine the resolvent sextic $f_{20}$ with $\theta$ as a root."

When I try to compute, for example

$\theta_1= x_1^2(x_2 x_5+ x_3 x_4)+ x_2^2(x_1x_3+ x_4x_5)+ x_3^2(x_1 x_5+ x_2x_4)+ x_4^2(x_1x_2+ x_3x_5)+ x_5^2(x_1 x_4+ x_2x_3)$

in elementary symmetric polynomials using substitution I end up with,

$\theta_1=\theta_1$

In the process I systematically replaced $x_i x_j$...with a function of $s_k$, expanded, collected terms and simplified.

One such substitution I used was,

$x_2 x_5+ x_3 x_4 = s_2-( x_1 x_2+ x_1 x_3+ x_2 x_3+ x_1 x_4+ x_2 x_4+ x_1 x_5+ x_3 x_5+ x_4 x_5)$

and another

$x_1^3(x_2 + x_3+ x_4+ x_5)= x_1^3(s_1- x_1)$

as well as

$x_1^4+ x_2^4+ x_3^4 +x_4^4+ x_5^4= s_1^4 - 4 s_1^2 s_2 + 2 s_2^2 + 4 s_1 s_3 - 4 s_4 $ .... and so on.

Since I have not been able to obtain a solution to the task, how should I be "computing the elementary symmetric functions of the $\theta_1 $ " ?

Answer 1

$\theta_1$ is not a symmetric polynomial under permutation of the $x_i$ (for example, if you swap $x_1$ with $x_2$, $\theta_1$ turns into something different), so you cannot express it as a polynomial in the elementary symmetric polynomials of the $x_i$.

Your quote does not talk about the $\theta_j$ individually, but about the elementary symmetric functions of the $\theta_j$, like for example, $\theta_1 + \theta_2 + \cdots + \theta_6$. This one is fully symmetric under permutation of the $x_i$, so it can be expressed in terms of the elementary symmetric polynomials of the $x_i$