TITLE: Primes in Gaussian Integers
QUESTION [7 upvotes]: Let $p$ be a rational prime. It is is well known that if $p\equiv 3\;\;mod\;4$, then $p$ is inert in the ring of gaussian integers $G$, that is, $p$ is a gaussian prime. If $p\equiv 1\;mod\;4$ then $p$ is  decomposed  in $G$, that is, $p=\pi_1\pi_2$ where $\pi_1$ and $pi_2$ are gaussian primes not  associated. The rational prime $2$ ramifies in $G$, that is $2=u\pi^2$, where $u$ is a unit in $G$ and $\pi$ a prime in $G$. 
where can I find a proof of this fact?  I want a direct proof, not a proof for the quadratic integers and then deduce this as a particular case.

REPLY [7 votes]: There are several places where you can find a direct proof of this. For instance, you can find it in the first 4 pages of Jurgen Neukirch's Algebraic Number Theory about the Gaussian integers. Also, LeVeque's Elementary Theory of Numbers has a short chapter dedicated to the Gaussian integers, where he proves this fact (see section 6.5).