Is ring $\mathbb{Z}_5[x]/I$ is a field, when: $I=(3x^3+2x+1)\mathbb{Z}_5[x]$?

Question

Is ring $\mathbb{Z}_5[x]/I$ is field, when: $I=(3x^3+2x+1)\mathbb{Z}_5[x]$?

Can you show me a step by step instruction how to solve this problem?

Answer 1

Since $\mathbb{Z}_5$ is a field, we know $\mathbb{Z}_5[x]$ is a PID. Note that $3x^3+2x+1$ is irreducible over $\mathbb{Z}_5$, since the polynomial is of degree $3$ and plugging in every element of $\mathbb{Z}_5$ does not yield a zero. This argument works because any polynomial of degree $2$ or $3$ over a field is irreducible if and only if it does not have a root in that field. Therefore the ideal generated by $3x^3+2x+1$ is prime, hence is maximal (we are over a PID). We thus conclude that the quotient is a field (quotient of a ring by a maximal ideal is a field).