Let $S_{2}$ act on $K[x,y]$ by sending $x$ to $y$ and $y$ to $x$. Let $A$ be the subring of elements of $K[x,y]$ fixed by $S_{2}$ i.e. $f \in A \iff f(x,y)=f(y,x)$. Find elements in $K[x,y]$ which generates it as an $A$-module.
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Related: http://math.stackexchange.com/questions/1004341/ring-of-polynomials-as-a-module-over-symmetric-polynomials/1004723#1004723 – user26857 Sep 20 '15 at 18:17
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$A=K[x,y]^{S_2}$ is the ring of symmetric polynomials in two variables, that is, $A=K[x+y,xy]$. Now I think you can find the desired element(s).
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