Is $(X^2+1)$ prime ideal in $\mathbb{R}[X,Y]$?

Question

The title already states my question: Is $(X^2+1)$ a prime ideal in $\mathbb{R}[X,Y]$?

I know that if this is true, Then $\mathbb{R}[X,Y]/(X^2+1)$ must be a domain. And if that is a domain then $(X^2+1)$ must be the kernel of a homomorphism from $\mathbb{R}[X,Y]$ to itself.

However, how do I prove that $(X^2+1)$ is the kernel of such homomorphism, or that this does not exist?

Thanks!

Answer 1

We have \begin{align*} {\bf{R}}[X,Y]/(X^{2}+1)\cong({\bf{R}}[X])[Y]/(X^{2}+1)\cong({\bf{R}}[X]/(X^{2}+1))[Y]\cong{\bf{C}}[Y]. \end{align*}