I am trying to identify the ring $$\mathbb{Z}[X]/(2,x^2+1).$$
Since I know that $2\equiv x^2+1 \equiv 0 \ mod \ (2,x^2+1)$, could I identify this as $Z_2[i]$ or it does not have anything to do with this? If it is correct, how could be a good way for me to justify it?
Thank you!
You are right: this is $R=\Bbb Z_2[i]=\{a+bi:a,b\in\Bbb Z_2\}$ where $i^2=-1$. One can say more about this. As we are in characteristic $2$ then $-1=1$, so $i^2=1$. Then $(1+i)^2=1+2i+1=0$ so is a nilpotent. Setting $u=1_i$, then $R=\Bbb Z_2[u]\cong \Bbb Z_2[X]/(X^2)$ where $u^2=0$.