Classification of nonzero prime ideals of $\mathbb{Z}[i]$ I know the classification of Gaussian primes: let $u$ be a unit of $\mathbb{Z}[i]$. Then the following are all Gaussian primes:
1) $u(1+i)$
2) $u(a+ib)$ where $a^2+b^2=p$ for some prime number p congruent to $1$ modulo $4$
3) $uq$ where $q$ is a prime number congruent to $3$ modulo $4$.
If i knew that $\mathbb{Z}[i]$ is a Principal ideal domain, i could say: a nonzero ideal $P$ of $\mathbb{Z}[i]$ is prime iff it is principal prime, say $P=p\mathbb{Z}[i]$ iff $p$ is a Gaussian prime, and conclude that every nonzero prime ideal $P$ of $\mathbb{Z}[i]$ belongs to one of the following families:
1) $P=(1+i)\mathbb{Z}[i]$
2) $P=(a+ib)\mathbb{Z}[i]$, where $a,b\in\mathbb{Z}$ and $a^2+b^2=p$ is an odd prime congruent to $1$ modulo $4$
3) $P=q\mathbb{Z}[i]$ where $q$ is an odd prime congruent to $3$ modulo $4$.
Now, suppose that for some reason i don't know that $\mathbb{Z}[i]$ is a PID. Can i still prove the classification above? I can't say an ideal is prime iff its generator is prime, so, how could i proceed in this case?
 A: To prove this result, you must prove along the way that $\mathbb{Z}[i]$ is a PID, to guarantee that you have not missed any non-principal ideals.  One cheap way to get around your restriction is to implicitly use the implication
$$\text{Euclidean domain} \Longrightarrow \text{PID}.$$
In the case at hand, we obtain a Euclidean algorithm on $\mathbb{Z}[i]$ as follows.  Given non-zero $x,y \in \mathbb{Z}[i]$, let $z \in \mathbb{Z}[i]$ denote the lattice point nearest $x/y$.  Geometrically, it's clear that 
$$\vert z - x/y \vert \leq 1/\sqrt{2},$$ in which the metric above is the standard metric on the plane.  Then $x-yz \in \mathbb{Z}[i]$ satisfies $\vert z-xy \vert < \vert y \vert$, so that $\vert \cdot \vert$ defines a Euclidean function on $\mathbb{Z}[i]$.  As in the case with the integers, a Euclidean algorithm can be used to find GCD (up to units).
Now, suppose that $\mathfrak{p}=(a,b) \subset \mathbb{Z}[i]$ is a prime ideal.  If $c$ denotes any GCD of $a$ and $b$, we have $c=(a,b)$.  Induction (and the fact that integer rings are Noetherian) implies that all ideals are principal.  For your problem, it would then suffice to classify all the principal ideals in $\mathbb{Z}[i]$, which is standard practice (look at norms).
Note: while this approach seems contrary to the question asked, I nevertheless believe that the best way to classify the prime ideals in $\mathbb{Z}[i]$ is to first prove that such ideals must be principal.  Other techniques would simply reprove this result, without the benefit of machinery.
Edit: This edit includes a proof of the fact that $\mathbb{Z}[i]$ is Noetherian.  Let $$\mathfrak{p}_1 \subsetneq \mathfrak{p}_2 \subsetneq \ldots $$
be an infinite chain of ideals in $\mathbb{Z}[i]$.  We note that $a_k:=\#\mathbb{Z}[i]/\mathfrak{p}_k$ is finite for all $k$, because both $\mathbb{Z}[i]$ and $\mathfrak{p}_k$ are abelian groups of rank $2$.  The sequence of positive integers $\{a_k\}$ is decreasing and does not stabilize, a contradiction.  Thus no such chains exist.
