If $p\in\mathbb{Z}$ is a prime and $p\equiv 1$ mod $4$, show that the quotient ring $\mathbb{Z}[i]/(p)$ has order $p^2$.

Question

The Problem: Let $p\in\mathbb{Z}$ be a prime with $p\equiv 1$ mod $4$. Show that $\mathbb{Z}[i]/(p)$ has order $p^2$.

Source: Abstract Algebra $\mathit{3^{rd}}$ edition by Dummit and Foote.

My Attempt: Indeed, $p=a^2+b^2$ for some $a, b\in\mathbb{Z}$, and we have that $N(p)=p^2$. I tried to establish an equivalence relation using the Euclidean Algorithm $\alpha=\beta p+\gamma$ with $N(\gamma)<N(p)=p^2$ for every $\alpha\in\mathbb{Z}[i]$, to no avail. Any HINT would be greatly appreciated.