I'm asked to prove that every symmetric polynomial $p(x,y)\in\mathbb{Q}[x,y]$ can be written as a polynomial in $xy$ and $x+y$.
I first want to show that if $p$ contains a term in the form $ax^iy^j$ where $i\neq j$ that it must also contain a term $ax^jy^i$ which I'm fairly confident I can do because $p$ needs to be symmetric. Then I can write any of these terms in the form $a(x^iy^j+x^jy^i)$ (and if $i=j$ we have $ax^iy^j$).
Now I'm unsure of my next step. I see that $x^iy^j+x^jy^i=x^iy^i(x^{j-i}+y^{j-i})$, and that $x^iy^i=(xy)^i$ but I'm unsure how to express $(x^{j-i}+y^{j-i})$ as a polynomial over $xy$ and $x+y$.
Thanks!