The Split(a, x) function computes the complete factorization of the univariate polynomial a in x.

If the third argument b is specified, it is assigned a set of elements $\{\mathrm{t1},...,\mathrm{tm}\}$ such that the polynomial a splits over $K=Q\left(\mathrm{t1},...,\mathrm{tm}\right)$, where Q denotes the field of rational numbers.

This function is part of the PolynomialTools package, and so it can be used in the form Split(..) only after executing the command with(PolynomialTools). However, it can always be accessed through the long form of the command by using PolynomialTools[Split](..).

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

$\mathrm{Split}\left(x^2+x+1\,x\right)$

$\left(x-\mathrm{RootOf}\left({\mathrm{\_Z}}^2+\mathrm{\_Z}+1\right)\right)\left(x+1+\mathrm{RootOf}\left({\mathrm{\_Z}}^2+\mathrm{\_Z}+1\right)\right)$

$\mathrm{Split}\left(x^2+yx+1+y^2\,x\,'b'\right)$

$\left(x+y+\mathrm{RootOf}\left({\mathrm{\_Z}}^2+y\mathrm{\_Z}+y^2+1\right)\right)\left(x-\mathrm{RootOf}\left({\mathrm{\_Z}}^2+y\mathrm{\_Z}+y^2+1\right)\right)$

$b$

$\left\{\mathrm{RootOf}\left({\mathrm{\_Z}}^2+y\mathrm{\_Z}+y^2+1\right)\right\}$