It is known that the field of multisymmetric rational functions (over a field of characteristic $0$), that is, 
rational function in variables $x\_{11}, \ldots, x\_{1m}, \ldots, x\_{n1}, \ldots, x\_{nm}$ that are invariant under the $S\_n$ action $g.x\_{ij}=x\_{g(i)j}$ is rational of transcendent degree $nm$. In the language of geometry $Sym^n(\mathbb{P}^m)$ is birationaly isomorphic to $\mathbb{P}^{nm}$. 
One example of a transcendent basis is elementary multisymmetric polynomials of multi-indices $(k, 0, \ldots, 0), k=1, \ldots n$ and $(k, 0, \ldots, 1, \ldots, 0), k=0, \ldots n-1$ and the $1$ is in places $2, \ldots, m$ (Discriminants, Resultants, and Multidimensional Determinants, Trm 2.8)


I am interested to know if there is a transcendent basis made entirely from multisymmetric power sum polynomials?



Transcendent basis for the field of multisymmetric functions
The answer is yes.


Indeed, Remark 7 of the paper 


Paul Görlach, Cordian Riener, Tillmann Weißer. Deciding positivity of multisymmetric polynomials. Journal of Symbolic Computation 74 (2016), 603-616.
(<https://www.sciencedirect.com/science/article/abs/pii/S074771711500098X>, see also <https://arxiv.org/abs/1409.2707>)


says that if we consider degrees arising from different $m$-tuples $(w\_1,\ldots,w\_m)$ of nonnegative integer weights, each multisymmetric polynomial $f$ can be written as a polynomial expression in the power sums for which the $w$-degree does not exceed the $w$-degree of $f$ for *all* weights $w$. Thus, for the elementary symmetric functions of the first type we are looking for power sums of multidegrees $(a\_1,\ldots,a\_m)$ with
 $$
a\_1w\_1+\cdots+a\_mw\_m\le kw\_1\le nw\_1
 $$
for all possible weights, and for the elementary symmetric functions of the second type we are looking for power sums of multidegrees $(a\_1,\ldots,a\_m)$ with
 $$
a\_1w\_1+\cdots+a\_mw\_m\le kw\_1+w\_p\le (n-1)w\_1+w\_p
 $$
for $p=2,\ldots,m$. Clearly, 
 $$
a\_1w\_1+\cdots+a\_mw\_m\le kw\_1\le nw\_1
 $$
for all nonnegative weights implies $a\_1\le n$, $a\_i=0$ for $i>1$, and, for a given $p$ between $2$ and $m$, 
 $$
a\_1w\_1+\cdots+a\_mw\_m\le kw\_1+w\_p\le (n-1)w\_1+w\_p
 $$
for all nonnegative weights implies $a\_1\le n-1$, $a\_p\le 1$, $a\_i=0$ for $i\ne 1,p$. So the constraints from Theorem 2.8' of Gelfand, Kapranov and Zelevinsky are exactly the same as constraints for the requested power sums, giving a required transcendence basis.


