Isomorphism between $\mathbb{Z}[i]/{p}$ and $\mathbb{F}^2_p$

Question

Say $p$ is a prime number such that $p=4k+1$. I need to prove that $\mathbb{Z}[i]/p$ (quotient ring) and $\mathbb{F}^2_{p}$ are isomorphic. I know that since $p \equiv 1 \mod4$ I can say that $p=xy$ where $x,y$ are conjugate prime numbers in $\mathbb{Z}[i]$, so I tried using the CRT (Chinese remainder theorem) - so I know that $\mathbb{Z}[i]/p$ is isomorphic to $\mathbb{Z}[i]/x\times\mathbb{Z}[i]/y$, but I wasn't able to take it from there. Any hint would be helpful.

Answer 1

More precisely we have $$\mathbb{Z}[i]/(p) \cong \mathbb{F}_p[X]/(X^2+1) \cong \begin{cases} \mathbb{F}_{p^2} &\text{if } p \equiv 3 \pmod 4\\ \mathbb{F}_{p} \times \mathbb{F}_{p} &\text{if } p \equiv 1 \pmod 4 \end{cases}$$

So the comment on whether it is written $\Bbb F_{p^2}$ or $\Bbb F_p^2$ was really meaningful.

Reference: Quotient ring of Gaussian integers $\mathbb{Z}[i]/(a+bi)$ when $a$ and $b$ are NOT coprime