Some subfield of ring $\mathbb{R}[x,y]/(x^2+y^2+1)$ Consider the ring $\mathbb{R}[x,y]/(x^2+y^2+1)$. 
The ideal $(x,x^2+y^2+1)/(x^2+y^2+1)$ is maximal, since modulo this ideal we get $\mathbb{R}[y]/(y^2+1)\cong \mathbb{C}$, a field. 
So $\mathbb{C}$ is a quotient field of the given ring $\mathbb{R}[x,y]/(x^2+y^2+1)$. 
I was wondering, whether $\mathbb{C}$ can be embedded in this ring? 

Q. Is there injective ring homomorphism from $\mathbb{C}$ into  $\mathbb{R}[x,y]/(x^2+y^2+1)$?


As an $\mathbb R$-vector space we know that $\mathbb R[x,y]/(x^2+y^2+1)$ has basis $x^m,x^ny$ for $m,n\geq0$. Thus every element can be written uniquely as $a+by$ with $a,b\in\mathbb R[x]$.
Now compute $(a+by)^2+1=(a^2-b^2(1+x^2)+1)+2aby$. If this vanishes, then $ab=0$, so either $a=0$ or $b=0$. Since the first term also vanishes, we must have either $a^2+1=0$ or $b^2(1+x^2)=1$. These are equations in $\mathbb R[x]$, which have no solutions. Thus there is no element $x\in\mathbb R[x,y]/(x^2+y^2+1)$ satisfying $x^2+1=0$, and hence no homomorphism $\mathbb C\to\mathbb R[x,y]/(x^2+y^2+1)$.
