What is the structure of $\mathbb Z[i]/\mathfrak p$ where $\mathfrak p$ is a prime.

Question

Initially, I was trying to prove both the isomorphism of $\mathbb Z[i]/\mathfrak p\cong \mathbb{Z}[x]/(p,x^2+1)\cong \mathbb{F}_p[x]/(x^2+1)$, where $\mathfrak p$ is a prime in $\mathbb Z[i]$ for some $p $ prime in $\Bbb Z.$ (Later on from the comment I got that the first isomorphism is not true for all prime in integers. So I am trying to ask, What is the structure of $\mathbb Z[i]/\mathfrak p$ where $\mathfrak p$ is a prime?

I have proved that for $ \phi: R \to S$ homomorphism the inverse image of a prime is prime. Now $\Bbb Z$ has inclusion in $\Bbb Z[i]$ that $i: \Bbb Z \to \Bbb Z[i]$ and hence the inverse, $i^{-1}(\mathfrak p)$ in $\mathbb Z$ is a prime and I can assume that $<p>$ is the corresponding prime ideal in $\Bbb Z$. I also know that $\mathbb Z[i]$ is a ED and hence $\mathfrak p=<P>$ where $P=a+bi$ a prime in $\mathbb Z[i]$ and also the norm of $P$, $N(P)=a^2+b^2$ is a prime. Am I going correct? Let me know if there is any other way.

So, I have a lot of pieces but I can't bring that together and for the 2nd isomorphism If I just consider the map $\psi:\mathbb{Z}[x]/(x^2+1)\to \mathbb{F}_p[x]/(x^2+1)$ s.t $f(x)\mapsto f(x)(\mod p)$ and then calculate the kernel, would that be enough? It is my begining in algebra, so I wan't check every details minutely. Any help would be appreciated. Even if I get some stepwise hint in stead of answer, that will help me in learning!

Edit From leoli's comment I can see that $\mathbb Z[i]/\mathfrak p$ can be a field as well. How to prove all the cases from the scratch as I am not getting any idea for that.

Answer 1

Let $\mathfrak p=(a+bi)$. There are two cases:

1. $ab=0$. In this case $\mathfrak p=(0)$, or $\mathfrak p=(p)$ with $p\in\mathbb Z$ a prime number, $p=4k+3$. Obviously, $\mathbb Z[i]/(0)=\mathbb Z[i]$. On the other side, $\mathbb Z[i]/(p)\simeq\mathbb F_p[X]/(X^2+1)$ which is a field with $p^2$ elements.

2. $ab\ne0$. In this case $p=a^2+b^2$ is a prime number. It follows that $\gcd(a,b)=1$ and we have $\mathbb Z[i]/\mathfrak p\simeq\mathbb F_p$. (For the last isomorphism see here.)

Answer 2

Hint : Consider the morphism \begin{align*} \varphi : \quad& \mathbb{Z}[i] \to \mathbb{F}_p[x]/(x^2+1)\\ & a+ib \mapsto \overline{\bar{a}+\bar{b}x}. \end{align*}

(where $\bar{a}$ and $\bar{b}$ denote the projections of the integers $a$ and $b$ in $\mathbb{F}_p$, and the big overline denotes the projection of $\bar{a}+\bar{b}x$ in the quotient).

Prove that this morphism is surjective, and that its kernel is exactly $(p)$. Then apply the first isomorphism theorem to get that $$\mathbb{Z}[i]/(p) \simeq \mathbb{F}_p[x]/(x^2+1) $$