I'm making a research on Galois theory, and found something interesting regarding the ring of symmetric polynomials:
At least up to 5 variables, we can rewrite the elementary symmetric polynomials with powers of symmetric polynomials of degree 1.
For the formulation of the question, i will introduce the index set $I\_q=\mathbb{N}\_{<q}$, where $\mathbb{N}$ are the natural numbers, including zero.


First, we define the subset $S\_k\subset I\_q^k$ as
$$S\_k=\{(i\_1;\dots;i\_k)\in I\_q^k\mid i\_1<i\_2<\dots<i\_k\}$$
This allows us to define the $k$-th elementary symmetric polynomial with $q$ variables $e\_k=e\_k(x\_0;\dots;x\_{q-1})$ as:
$$e\_k=\sum\_{(i\_1;\dots;i\_k)\in S\_k}x\_{i\_1}\dots x\_{i\_k}$$
We also define the symmetric $p$-th powers of first degree polynomials $s\_j^p=s\_j^p(x\_0;\dots;x\_{q-1})$ to be:
$$s\_j^p=\sum\_{(i\_1;\dots;i\_j)\in S\_j}(x\_{i\_1}+\dots+x\_{i\_j})^p$$
The question is then: can every elementary symmetric polynomial $e\_k$ be written as a linear combination of these symmetric $k$-th powers of first degree polynomials $s\_j^k$ for an arbitrary number of variables $q$?


As said earlier, i have made some progress, and found a pattern up to $q=5$ variables. I will leave my findings here:


[For $q=1$]


$$e\_1=s\_1^1$$


[For $q=2$]


$$\begin{align\*}
e\_1&=s\_1^1+0s\_2^1\\
e\_2&=\frac{1}{2}(-s\_1^2+s\_2^2)
\end{align\*}$$


[For $q=3$]


$$\begin{align\*}
e\_1&=s\_1^1+0s\_2^1+0s\_3^1\\
e\_2&=\frac{1}{2}(-s\_1^2+0s\_2^2+s\_3^2)\\
e\_3&=\frac{1}{6}(s\_1^3-s\_2^3+s\_3^3)
\end{align\*}$$
Continuing in a similar fashion, we can construct $q$-dimensional vector spaces $V\_k$ over $\mathbb{Q}$, spanned by the set of symmetric $k$-th powers $G\_k=\{s\_1^k; s\_2^k;\dots;s\_q^k\}$.
The $k$-th elementary symmetric polynomial is then a vector $\vec{e\_k}\in V\_k$ in this setting. Note that these representations are not unique, e.g., for $q=2$, $\vec{e\_1}=(1;0)=(0;1)$, and hence $G\_k$ is not a basis for $V\_k$. I'll add a few more values for $q$:


[For $q=4$]


$$\begin{align\*}
\vec{e\_1}&=(1;0;0;0)\\
\vec{e\_2}&=\frac{1}{2}(-1;0;0;1)\\
\vec{e\_3}&=\frac{1}{6}(4;2;-3;-1)\\
\vec{e\_4}&=\frac{1}{24}(-1;-1;1;1)
\end{align\*}$$


[For $q=5$]


$$\begin{align\*}
\vec{e\_1}&=(1;0;0;0;0)\\
\vec{e\_2}&=\frac{1}{2}(-1;0;0;0;1)\\
\vec{e\_3}&=\frac{1}{6}(-6;0;-1;3;0)\\
\vec{e\_4}&=\frac{1}{24}(-5;2;-3;4;-1)\\
\vec{e\_5}&=\frac{1}{120}(1;-1;1;-1;1)
\end{align\*}$$


etc.



Can the ring of symmetric polynomials be generated by powers of symmetric polynomials of degree 1?
Yes.


For variables $x\_1,\dots, x\_k$, we have


$$ \sum\_{J \subseteq \{1,\dots, k \} } (-1)^{ k- |J|} \left(\sum\_{j \in J} x\_j\right)^k = k! x\_1 \dots x\_k $$


so that
$$ e\_k = \sum\_{(i\_1; \dots; i\_k )\in S} x\_{i\_1} \dots x\_{s\_k} = \frac{1}{k!} \sum\_{(i\_1; \dots; i\_k )\in S} \sum\_{J \subseteq \{i\_1,\dots, i\_k \}}(-1)^{ k- |J|} \left(\sum\_{j \in J} x\_{j} \right)^k $$


and each set $J$ occurs $\binom{ q-|J|}{k- |J|} $ times, so


$$ e\_k = \sum\_{ \substack{ J\subseteq I\_q \\ |J| \leq k}} (-1)^{ k -|J|} \frac{ \binom{ q- |J|} {k-|J|}}{k!} \left(\sum\_{j \in J} x\_j\right)^k =\sum\_{j=0}^k (-1)^{ k -j} \frac{ \binom{ q- j} {k-j}}{k!} s\_j ^k.$$


