Let $\mathbb{Z}[i]$ be the rings of Gaussian integers. Then which of the following statements are true:
a) if $P$ is a prime ideal of $\mathbb{Z}[i]$, then $\mathbb{Z}[i]/P$ is a field.
b) if $P$ is a prime ideal of $\mathbb{Z}[i]$, then $\mathbb{Z}[i]/P$ is always a degree 2 extension of its prime fields.
c) for any prime $p \in \mathbb{Z}$, the ideal $P$ generated by "$p$" in $\mathbb{Z}[i]$ is a prime ideal.
d) for any non zero prime ideal $P$ in $\mathbb{Z}[i]$, the intersection $\mathbb{Z} \cap P$ is a non zero ideal of $\mathbb{Z}$.
I think option (c) is coprrect
a) Consider $P=(0)$
b) Consider $P=(0)$
c) Try $p=2$ and consider what "prime ideal" would imply for $(1+i)\cdot (1-i)=2$.
d) Let $a+bi\in P\setminus\{0\}$. What can you say about $(a+bi)\cdot(a-bi)$?