Isomorphism Between Gaussian Integers Modulo Some Element and the Ring of Integers Modulo it's Norm.

Question

Show that for $\theta\in\mathbb{Z}[i]$, if $\theta=a+bi$ for $\mathrm{gcd}(a,b)=1$, we have $\mathbb{Z}[i] / (\theta)\cong\mathbb{Z}/{(N(\theta))}.$

Answer 1

Consider the natural ring homomorphism $\varphi:\mathbb Z\to\frac{\mathbb Z[i]}{\theta\mathbb Z[i]}$. Since $\gcd(a,b)=1$, there exist $r,s\in\mathbb Z$ such that $ar+bs=1$, which implies $(s+ri)\theta=as-br+i$, and so $i+\theta\mathbb Z[i]$ is in the image of $\varphi$. Consequently $\varphi$ is surjective.

Now if $n\in N(\theta)\mathbb Z$ then $n=mN(\theta)=m(a-bi)\theta$ for some $m\in\mathbb Z$, which implies $n\in\ker(\varphi)$. Conversely, if $n=(c+di)\theta$ for some $c,d\in\mathbb Z$ then $n(a-bi)=N(\theta)(c+di)$, hence $na=N(\theta)c$, that is $N(\theta)$ divides $na$ in $\mathbb Z$; since $\gcd\bigl(N(\theta),a\bigr)=\gcd(a^2+b^2,a)=1$ because $\gcd(a,b)=1$ then $N(\theta)$ divides $n$ in $\mathbb Z$. This shows that $\ker(\varphi)=N(\theta)\mathbb Z$. Now apply the isomorphism theorem.