Calculations in polynomial quotient rings e.g. $\mathbb{Z}[X]/(X^2+1)$ Is there a nice way to perform calculations within a quotient ring? for instance, an exercise I had was to show $\mathbb{Z}[X]/(X^2+1) \cong \mathbb{Z}[i]$, but this got me thinking; If I didn't know this, how could I calculate in $\mathbb{Z}[X]/(X^2+1)$ easily?  (and of course any other general quotient ring)? 

In $\mathbb{Z}[x]/(x^2+1)$ you can simply calculate as in $\mathbb{Z}[x]$ on representants of equivalence classes and then, if you need it, reduce them applying polynomial division by $x^2+1$ and keeping the remainder. The only "complicated" operation is to check if two elements $p(x)$ and $q(x)$ and  are equal, for which you must show (again using polynomial division) that
$$p(x)-q(x)\equiv 0\mod{x^2+1}$$
