Factoring Gaussian integers

Question

How do I factor the elements $2, 3$ and $5$ of the ring $\mathbb{Z}[i]$? Are they not primes, that is $ 2=2 \times 1$, etc? (an exercise from Vinberg's Algebra).

Answer 1

Being a prime depends very much in what ring we are working. So for instance $2$ and $5$ are primes in $\mathbb{Z}$ while they are composites in $\mathbb{Z}[i]$ the Gaussian integers.

One has the following theorem.

An odd prime number $p\in\mathbb{Z}$ is composite in the Gaussian integers $\mathbb{Z}[i]$ if and only $p=4k+1$ for some integer $k$.

Answer 2

Hint: Think about both $a+bi$ and $a-bi$ for some small values of $a$ and $b$.