Gaussian prime

A Gaussian prime is a nonzero nonunit Gaussian integer irreducible in $\mathbb{Z}[i]$. Since this ring is a Euclidean domain, irreducible and prime elements coincide. For $a,b\ne 0$, $a+bi$ is a Gaussian prime exactly when $a^2+b^2$ is an ordinary prime number. On a coordinate axis, it is prime exactly when the nonzero coordinate has absolute value an ordinary prime congruent to three modulo four.

Thus $1+i$, $2+i$, and $3$ are Gaussian primes, while $2=-i(1+i{)}^2$ and $5=(2+i)(2-i)$ are not. The multiplicative norm $N(a+bi)=a^2+b^2$ makes these factorizations and primality tests transparent.