Where is wrong with this fake proof that Gaussian integer is a field?
"Since $x^2+1$ is an irreducible element, the ideal $(x^2+1)$ is maximal"
Is this true in a generic integral domain? Consider the ring $Z[x,y].$ We have that $x$ is an irreducible element, but $(x)$ is not a maximal ideal, as it is contained in the ideal $(x,y)$ which is still not the entire ring.
$(x^2+1)$ is a prime ideal but not maximal.
it happens in a ring of Krull-dimension $\geq 2$. $\dim \mathbb{Z}[X] = 2$.