How to equip $\frac{\mathbb{Z}}{5\mathbb{Z}}$ with the structure of a $(\frac{\mathbb{Z}}{5\mathbb{Z}}[x])/(x^2+1)$-module?
I know that $x^2+1=(x-2)(x-3)$ in this field, so that makes it a reducible polynomial. So this makes this unfamiliar to me.
I think I should start by defining the multiplication. For instance, what does $x*1=$?
For $(x-2)(x-3)=0$, I should have that $x=2$ or $x=3$.
Do I just choose one and roll with it?