TITLE: Idempotents in $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i]$
QUESTION [7 upvotes]: Letting $\mathbb{Z}[i]=\left\{a+bi:a,b \in \mathbb{Z} \right\}$ be the ring of Gaussian integers, how many idempotents are there in $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i]$? 

I came across this in my reading, and I'm a little unsure of how to approach this. Any help would be appreciated!

REPLY [6 votes]: We have $$\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i]\cong\Bbb Z[i][X]/(X^2+1).$$ Every element of the ring $\Bbb Z[i][X]/(X^2+1)$ can be uniquely written as $a+bx$, with $x^2=-1$. If $(a+bx)^2=a+bx$, then $a^2-b^2=a$ and $2ab=b$. Now we have two cases: $b=0$ and $a=0,1$ or $b\ne 0$ and then $a=1/2\notin\Bbb Z$. It follows that $0,1$ are the only idempotents.