### Compute elementary Symmetrical functions on roots of a polynomial

~~If ~~`u0,u1,...,u4`

are variables defined in `R=PolynomialRing(QQ,5,'u')`

and `RR.<T>=R[]`

is the ring where I defined a polynomial `(T-u0)*(T-u1)*...*(T-u4)`

, one can How to compute the ~~elementary symmetrical ~~elementart symmetric functions ~~on these variables: namely something like ~~~~e[1,1](u0,...u4)~~

to get `u0u1+u0u2+...+u3u4`

.

In the documentation about `S=SymmetricFunctions(QQ);e=S.monomials()`

, you can rename variables by "alphabet", eg `v=e[1,1,1].expand(4,alphabet='u')`

and then coerce this in the ring, `R(v)`

.
To check, one may apply a permutation, say `p=SymmetricGroup(4).subgroup([(1,2)])`

to it, by `R(v)(*p[1](R.gens()))`

, which seems to work!

Now, if the `uie[1,1,..]`

's ~~are not ~~to calculate variables ~~but calculated ones, say ~~`a,b,c,...`

~~ then how to assign the ~~`e[1,1,..]`

to calculate teh elementary symmetrical functions , typically computing "elemenfs" on ~~these ~~`a,b,c,...`

?roots of a polynomial? This seems such a basic question, but I can't find anywhere.