The GIfactor function returns the Gaussian integer factorization of c.

The answer is in the form: $u{\left(f_1\right)}^{e_1}\cdot \dots \cdot {\left(f_n\right)}^{e_n}$ such that $c=u\cdot f_1^{e_1}\cdot \dots \cdot f_n^{e_n}$ where $u$ is one of four units, $f_1,...,f_{\mathrm{n}}$ are the distinct primary prime factors of c, and $e_1,...,e_{\mathrm{n}}$ are their multiplicities.

The number $x+\mathrm{I}y$ of the four associated primes $a+\mathrm{I}b$, $-a-\mathrm{I}b$, $-b+\mathrm{I}a$, $b-\mathrm{I}a$ is singled out as primary in which we have simultaneously either ($x=1\mathbf{mod}4\mathbf{and}y=0\mathbf{mod}4$) or ($x=\mathrm{-1}\mathbf{mod}4\mathbf{and}y=2\mathbf{mod}4$).

The expand function may be applied to cause the factors to be multiplied together again.

$\mathrm{with}\left(\mathrm{GaussInt}\right)\:$

$\mathrm{GIfactor}\left(-72-100\mathrm{I}\right)$

$\left(\mathrm{I}\right){\left(1+\mathrm{I}\right)}^4\left(\mathrm{-3}-2\mathrm{I}\right)\left(\mathrm{-3}+8\mathrm{I}\right)$

$\mathrm{expand}\left(\right)$

$\mathrm{-72}-100\mathrm{I}$

$\mathrm{GIfactor}\left(12101-15295\mathrm{I}\right)$

$\left(\mathrm{-1}\right)\left(1+\mathrm{I}\right)\left(1+4\mathrm{I}\right)\left(5+8\mathrm{I}\right)\left(9+16\mathrm{I}\right)\left(\mathrm{-7}-18\mathrm{I}\right)$

$\mathrm{expand}\left(\right)$

$12101-15295\mathrm{I}$