Q:

How to prove whether $x^{2}+y^{2}+1$ is irreducible over $\mathbb{C}$ or not?

Let's consider a 2-variable polynomial $f(x, y)= x^{2}+y^{2}+1$. It can be established that it's irreducible over $\mathbb{R}$. 
For example, if it's irreducible over $\mathbb{R}$ as a polynomial of two variables, then if we assume $y=1$ the polynomial must be irreducible too, whereas $x^{2}+2$ doesn't look so (By the Eisenstein's criterion, for example). 
The situation becomes slightly more interesting over $\mathbb{C}$. How to prove, in particular, that the $x^{2}+y^{2}+1$ is irreducible over $\mathbb{C}$ or not and which technique may be applied in order to cope with such kind of problems in  general cases?
Any piece of advice would be much appreciated.

A:

Consider the ring $R=\mathbb{C}[x]$, which is a PID. The polynomial $y^2+(x+i)(x-i)$ is irreducible in $R[y]$ by Eisenstein's criterion, as $x+i$ is irreducible in $R$ and $x-i$ doesn't divide $x+i$.
Note that Eisenstein's criterion applies to any PID, with the same proof as in the case for integers.

