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

The factorization is returned in the form $\left[u\,\left[\left[f_1\,m_1\right]\,\mathrm{...}\,\left[f_n\,m_n\right]\right]\right]$ such that $a=u{f_1}^{m_1}\mathrm{...}{f_n}^{m_n}$, where each $f_k$ is a unit normal irreducible polynomial and its multiplicity $m_k$ is a positive integer.

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 Splits(..) only after executing the command with(PolynomialTools). However, it can always be accessed through the long form of the command by using PolynomialTools[Splits](..).

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

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

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

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

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

$b$

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