Squarefree Gaussian integer

A nonzero Gaussian integer is squarefree if no square of a Gaussian prime divides it. Equivalently, every exponent in its Gaussian prime factorization is at most one, disregarding multiplication by units.

The Gaussian integer $2$ is not squarefree, because $2=-i(1+i{)}^2$. By contrast $5=(2+i)(2-i)$ is squarefree: its factors are nonassociate Gaussian primes. Thus squarefreeness in $\mathbb{Z}[i]$ need not agree with squarefreeness in $\mathbb{Z}$. This distinction is essential when using a Gaussian Möbius function to exclude repeated prime factors.