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

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?

• In general for a ring $R$ and a ideal $I$, when is $R/I$ a field? Jun 8, 2020 at 6:41

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).