Show that $\mathbb{Z}[x,y]$ is generated as a module over its subring $\mathbb{Z}[x+y,xy]$ by its elements $1,x$.
Could you give me a hint on this problem? I do not understand how this can be true, for example, how would you generate $y^{n}+\dots +y+1\in \mathbb{Z}[x,y]$ in the way described above?
I tried fixing $k$ and inducting over $n$ in the arbitrary polynomial $p=a_{n,k}x^{n}y^{k}+\dots +a_{0,0}\in \mathbb{Z}[x,y]$, but this did not seem to work because of the polynomial above.
Any help would be appreciated.
