Is $\mathbb{Z}[x]/(x^2 + x + 1, 9)$ isomorphic to $\mathbb{F}_{81}$?

Question

I'm trying to determine whether $\mathbb{Z}[x]/(x^2 + x + 1, 9)$ isomorphic to $\mathbb{F}_{81}$, but I'm confused since $\mathbb{Z}/9\mathbb{Z}$ is not a field, but $x^2 + x + 1$ is irreducible in $\mathbb{Z}/9\mathbb{Z}$. Thus the quotient in the title is a field, but which field?

Answer 1

In a commutative ring $R$ we have $\frac{R}{I}$ is a field if and only if $I$ is a maximal ideal.

When we have a field $F$ we have that $F[x]$ is a PID, so an ideal is maximal if and only if it is generated by an irreducible polynomial.

In this case however $\mathbb Z$ is not a field, so we cannot conclude $\mathbb Z[x]$ is a PID.

So we cannot conclude $(x^2+x+1,9)$ is a maximal ideal. in fact it is not, as $(3,x^2+x+1)$ contains it properly and is not all of $\mathbb Z[x]$.