TITLE: Where is wrong with this fake proof that Gaussian integer is a field?
QUESTION [21 upvotes]: The Gaussian integer $\mathbb{Z}[i]$ is an Euclidean domain that is not a field, since there is no inverse of $2$. So, where is wrong with the following proof?

Fake proof
First, note that $\mathbb{Z}[X]$ is a integral domain. Since $x^2+1$
  is an irreducible element in $\mathbb{Z}[X]$, the ideal $(x^2+1)$ is
  maximal, which implies $\mathbb{Z}[i]\simeq\mathbb{Z}[X]/(x^2+1)$ is a
  field.

REPLY [2 votes]: The statement that $(x^2+1)$ is maximal is false.
The maximal ideals of $\mathbb Z[x]$ are of the form $(p, x)$ where $p$ is a prime.