Proof this factor ring is field.

Question

How to prove that $\mathbb {Z}[i] /M $is field where $M=\{a+bi,:3|a,3|b\}$

I thought I would use a homomorphism to convert $ \mathbb{Z}[i] /M $ to $\mathbb{Z}/3\mathbb{Z} $ then I will prove $\mathbb {Z}/3\mathbb {Z}$ is field but I don't now how to conevert it.

Any hint for prove this factor ?

Thanks.

Answer 1

Hint : You need to show that $M$ is a maximal ideal. Note that $M$ is the ideal generated by $3$ and $\mathbb{Z}[i]$ is a PID. So it is enough to show that $3$ is a prime in $\mathbb{Z}[i]$.

Answer 2

$\mathfrak m$ is the ideal of $\mathbf Z[i]$ generated by $3$. Now, $\;\mathbf Z[i]\simeq \mathbf Z[X]/(X^2+1)$, so $$\mathbf Z[i]/(3)\simeq \Bigm/3\cdot \mathbf Z[X]/(x^2+1)\simeq \mathbf Z[X]/(3,X^2+1)\simeq (\mathbf Z/3\mathbf Z)[X]/(X^2+1),$$ so the question is:

Is $X^2+1$ irreducible over $\mathbf Z/3\mathbf Z\,$?