Primary Gaussian integer
= Primary Gaussian integer
{title2=$\alpha\equiv1\pmod{(1+i)^3}$}
An odd <Gaussian integer> is primary when it satisfies the displayed congruence. The four units have distinct images in $(\mathbb Z[i]/(1+i)^3)^\times$, a group of order four; hence every odd ideal has a unique primary generator. Indeed $v_{1+i}(-2)=2$ and $v_{1+i}(i-1)=v_{1+i}(-i-1)=1$.