How to find the prime ideals of $\mathbb{Z}[i]$ using the associated map of spectra? I want to know how do you use the map $$^a\varphi : \mathrm{Spec} \,\mathbb{Z}[i]\to \mathrm{Spec}\, \mathbb{Z}$$
that is given by $p \mapsto \varphi^{-1}({p})$, where $\varphi : \mathbb{Z} \to\mathbb{Z}[i]$ is the inclusion map, to find the 'points' in $\mathrm{Spec}\, \mathbb{Z}[i]$?
This question is an example in Shafarevich's Basic Algebraic Geometry (pg. 6). There the author writes that "$(^a\varphi)^{-1}(\{(p)\})$ is the set of prime ideals of $\mathbb{Z}[i]$ that divide $p$". I want to know how can I verify this?
I must be doing something very wrong because I've found that the set ought to contain the ideals that are divisible by $p$.
Also, how is this map any useful in finding the points in the spectrum of $\mathbb{Z}[i]$? How would you go about using this map to find, say, what $(^a\varphi)^{-1}(\{(17)\})$ is? This example also isn't clear as to how can this map guarantee that the prime ideals of $\mathbb{Z}[i]$ are only those that are in the sets $(^a\varphi)^{-1}(\{(p)\})$ for every prime number $p$.
 A: For any general ring map $f: B\to A$, let $f'$ denote the induced map on the Spec. Let $\mathfrak p$ be a prime ideal of $B$ and let us investigate what the prime ideals of $A$ that map to $\mathfrak p$ are.
That is, we want $\mathfrak q \subset A$ such that $f^{-1}(\mathfrak q) = \mathfrak p$. In particular, $f(\mathfrak p) \subset \mathfrak q$. Therefore, the primes $\mathfrak q$ that we want are certainly contained in the Spec of $A/f(\mathfrak q)A$. 
Next, we have an induced map $\overline f: B/\mathfrak p \to A/f(\mathfrak p)A$. Now, any element $\overline b \in B/\mathfrak p$ will certainly not map to any of the primes $\mathfrak q$ that we are looking for. That is to say, the primes we are looking for are contained in $(B/\mathfrak p)^{-1}(A/f(\mathfrak p)A) = A \otimes_B\kappa(B)$ where $\kappa(\mathfrak p)$ is the residue field at $\mathfrak p$.
In fact, a little more thought will show that the primes of $A \otimes_B\kappa(B)$ are exactly what we are looking for.
If you don't understand this answer, the first thing to do would be to make sure that you understand the correspondence between primes of $B$ containing $\mathfrak p$ and primes of $B/\mathfrak p$ and similarly, for any multiplicative set $S$, the primes of $S^{1}B$ and primes of $B$ avoiding $S$.
