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For some n>4, can one find two symmetric polynomials $S_1$ and $S_2$ in $\mathbb{Q}[x_1,...,x_n]$ such that $S_1+x_1S_2$ is a square in $\mathbb{Q}[x_1,...,x_n]$?

I have such a construction for the case $n \leq 4$, and my construction for n=4 itself is already quite tricky. I suspect there is no such construction for $n>4$, but I'm still struggling to find the best theoretical approach. Any suggestion?

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    $\begingroup$ One possible approach would be to consider this as a Galois-theoretic question about the $S_n$-extension $F={\mathbb Q}(x_1,...,x_n)/{\mathbb Q}(s_1,...,s_n)=K$, where $s_i$ are the elementary symmetric functions in the $x_i$. The question is whether there is an element of $K(x_1)$ which is a square and which is linear in $x_1$. I don't see how to do it for large $n$, but this might be a helpful framework. $\endgroup$ – Tim Dokchitser Mar 17 '11 at 3:36

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