a question about abstract algebra,the order of $\Bbb Z_{5}[x]/ (x^3+x+1)$

Question

Firstly, I have proven that $x^3+x+1$ is irreducible in $\Bbb Z_{5}[x]$,then how can I know the order of $$\Bbb Z_{5}[x]/ (x^3+x+1),$$ where $(x^3+x+1)$ means the ideal generated by $x^3+x+1$.

Can osmeone give me some help?

Answer 1

Note that polynomial long division tells us that for any polynomial $f(x)$, it can be uniquely written as $f(x) = q(x)(x^3+x+1) + r(x)$, where $r(x)$ is of degree 2 or less.

This says that the polynomials $ax^2 + bx+c$ form representatives for the equivalence classes in the quotient, allowing the order to be computed.

Answer 2

More generally, $$\left|\frac{\mathbb F_p[x]}{\text{ <an irreducible polynomial of degree $n$ over } \mathbb F_p>}\right|=p^n$$