4.II.11C

Let $R$ be the ring of Gaussian integers $\mathbb{Z}[i]$, where $i^{2}=-1$, which you may assume to be a unique factorization domain. Prove that every prime element of $R$ divides precisely one positive prime number in $\mathbb{Z}$. List, without proof, the prime elements of $R$, up to associates.

Let $p$ be a prime number in $\mathbb{Z}$. Prove that $R / p R$ has cardinality $p^{2}$. Prove that $R / 2 R$ is not a field. If $p \equiv 3 \bmod 4$, show that $R / p R$ is a field. If $p \equiv 1 \bmod 4$, decide whether $R / p R$ is a field or not, justifying your answer.

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