I want to find the prime ideals of $R=F[x,y]/(x^2+y^2+1)$, where $F=\mathbb{C}$ or $\mathbb{R}$.
Ok, $(x^2+y^2+1)$ is irreducible, then $(0)$ is a prime ideal in $R$. Also any other prime in $R$ must be prime in $F[x,y]$ and contain $(x^2+y^2+1)$; hence it won't be generated by a single irreducible polynomial. Then I have the ideals on the form $(q(x), f(x,y))$ left, where $q$ is irreducible in $F[x]$ and $f$ is irreducible in $F[x,y]/(q(x))$.
Which of them contains $(x^2+y^2+1)$?
I also tried to build a isomorphism $\varphi :R\rightarrow F[\sin (t)i, \cos (t)i]$ and see it like a two variable polynomial ring over $F$. Didn't work...
Thanks in advance!