In $\Bbb Z_3[x]$ let $f(x)=x^3+x^2+x+1$ and let $A=\Bbb Z_3[x]/(f) $ be the quotient ring. I'm asked to find the cardinality of $A $ and whether it is a field.
For the cardinality, I should find that $A $ is in bijection with the set of polynomials with degree smaller than $3$ (and in $\Bbb Z_3[X]$), hence $|A|=3^3$.The thing is, I can't really visualise $A$. I know its elements are the classes of the remainders of Euclidean division by $f$, but how do I get these?
For the other question, the given answer is that $f(x)=(x^2+1)(x+1) $ is reducible in $\Bbb Z_3[X]$, thus $A$ is not a field. Why does this work? Is this a general property of quotient rings of polynomials, i.e. if $a\in A[X]$ is reducible then $A[X]/(a)$ is not a field?