Let $\varphi : \mathbb{Z}[x,y] \to \mathbb{C}$ be a ring homomorphism. Then $\ker{\varphi}\subset \mathbb{Z}[x,y]$ is prime.
I'm assuming $\mathbb{Z}[x,y]$ a ring?
The argument I have is that $\mathbb{Z}[x,y]/\ker{\varphi}$ is an integral domain. I'm looking back at my notes and I see I've written that for $\operatorname{Im}(\varphi)\subset \mathbb{C}$, we can't say the image is a field because $\mathbb{C}$ is not a finite field. I'm confused because I thought any 'structured' subset of $\mathbb{C}$ could be a subfield.