I am given $L:=\frac{\mathbb{F}_3[x]}{(x^3+x+1)}$ and I have to prove different properties about this object.
First of all, since the polynomial for which I make the quotient is reductible $$x^3+x+1=(x-1)(x^2+x+2)$$ we have that $L$ is a ring and not a field, right?
Second, I get asked how many elements does $L$ have. If $L$ was a field, then I know that the number of elements in $L$ would be $3^3$, but since $L$ is not a field, so I suspect that my ring will have a different number of elements. How do I calculate this number?
Thirdly, I get asked to see if $L$ is an integral domain. For this I would like to have a list of my elements in $L$, but I struggle to get this list.
L is indeed not a field, because it is not a domain. You can see this from the factorisation of $f:= x^3+x+1$, because $\overline{f} = \overline{(x-1)} \cdot \overline{(x^2+x+2)}=\overline{0}$ where the overline stands for reducing polynomials in $\mathbf{F}_3[x]$ modulo $f$. Where $\overline{(x-1)}$, $\overline{(x^2+x+2)}$ are not zero in $L$.
Furthermore, $L$ is a ring generated by $3$ elements (i.e. $\overline{1}, \overline{x},\overline{x}^2$), because you can reduce modulo $f$ every polynomial of degree larger than three (or equal) to a combination of these.