A question about the action of $S_n$ on $K[x_1,...,x_n]$

Question

Let ${K}$be the field ($\,Char\,K\not=0)$. Let $n\in \mathbb{Z}^{+}$. $S_n$ acts on $K[x_1,...,x_n]$in the following way:

If $p\in K[x_1,...,x_n]$ and $\sigma\in S_n$, then $\sigma p$ is the polynomial $p(x_{\sigma(1)},x_{\sigma(2)},...,x_{\sigma (n)})$.

Question: Let $H$ be a subgroup of $S_n$, must there exist a polynomial $p\in K[x_1,...,x_n]$ such that $stab(p)=H$ ?

(Where $stab(p)$ is defined to be $\{\sigma\in S_n|\sigma p=p\})$

Thank you

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Edit: I also want to know the answer to the question in the case that our field $K$ is $\mathbb{C}$

Answer 1

Yes. Let $f(x_1,\ldots, x_n)=\prod_{k=1}^n x_k^k$. Then $p=\sum_{h\in H} h(f)$ has the desired property. (This works for all fields, also those of nonzero characteristic)