I am doing questions on polynomial rings, working out of Dummit and Foote. I am struggling on a question.
Let $\mathbb{C}[x+y,xy]$ be the polynomial ring in two variables $x+y$ and $xy$ over the complex numbers. Show that $\mathbb{C}[x+y,xy] \cong \mathbb{C}[z,w]$.
To prove this, there is a pretty obvious map $\varphi: \mathbb{C}[z,w] \rightarrow \mathbb{C}[x+y,xy]$ where we say $\varphi(z) = x+y$ and $\varphi(w) = xy$ and then extend the map $\varphi$ to make it into a ring homomorphism. Given a polynomial $p(x+y,xy)$, I just say that $\varphi(p(z,w)) = p(x+y,xy)$. So $\varphi$ is surjective. I do not know how to obtain injective. I let $\varphi(p(z,w)) = 0$, and the ring homomorphism implies that $p(x+y,xy) = 0$. I think that I need to show that $x+y$ and $xy$ are relatively prime in $\mathbb{C}[x,y]$, but I do not know if that is sufficient. I appreciate any help on this question. Thanks.