The GIroots function computes the Gaussian integer roots of a univariate polynomial over the Gaussian integer domain. The roots are returned as a list of pairs of the form $\[\[r_1,m_1\],\dots ,\[r_n,m_n\]\]$ where $r_i$ is a root of the polynomial a with multiplicity $m_i$, that is, ${\left(x-r_i\right)}^{m_i}$ divides a.

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

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

$\mathrm{GIroots}\left(\mathrm{expr}\right)$

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

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

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

$\mathrm{GIroots}\left(f\right)$

$\left[\left[1+\mathrm{I}\,1\right]\right]$