The function GIfacpoly computes the factorization of a multivariate polynomial with Gaussian integer coefficients.

The result is returned in the form $\[u,\[\[p_1,{\ⅇ}_1\],\mathrm{...},\[p_m,{\ⅇ}_m\]\]$ where $z=u{p_1}^{{\ⅇ}_1}...{p_m}^{{\ⅇ}_m}$, $p_i$ is an irreducible polynomial over Gaussian ring with the gcd of its coefficients being 1, $e_i$ is its exponent (multiplicity) and u is a Gaussian integer.

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

$\mathrm{expr1}≔x^4-17x^3-29\mathrm{I}{x}^3-188x^2+339\mathrm{I}{x}^2+1682x-86\mathrm{I}x-1178-1244\mathrm{I}$

$\mathrm{expr1}≔x^4-17x^3-29\mathrm{I}{x}^3-188x^2+339\mathrm{I}{x}^2+1682x-86\mathrm{I}x-1178-1244\mathrm{I}$

$\mathrm{GIfacpoly}\left(\mathrm{expr1}\right)$

$\left[1\,\left[\left[-x+1+\mathrm{I}\,1\right]\,\left[-x+7+11\mathrm{I}\,1\right]\,\left[-x+4+9\mathrm{I}\,1\right]\,\left[-x+5+8\mathrm{I}\,1\right]\right]\right]$

$\mathrm{expr2}≔\left(2+\mathrm{I}\right)x^2+\left(3+7\mathrm{I}\right)x+\left(6-14\mathrm{I}\right)$

$\mathrm{expr2}≔\left(2+\mathrm{I}\right)x^2+\left(3+7\mathrm{I}\right)x+6-14\mathrm{I}$

$\mathrm{GIfacpoly}\left(\mathrm{expr2}\right)$

$\left[\mathrm{-I}\,\left[\left[-x+1+\mathrm{I}\,1\right]\,\left[x-2\mathrm{I}x+10-4\mathrm{I}\,1\right]\right]\right]$