Find elements in $K[x,y]$ which generates it as an $A$-module. 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. 

$A=K[x,y]^{S_2}$ 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).
