This function computes square-free factorizations of polynomials and rational functions over algebraic function fields or algebraic number fields. The square-free factorization is returned in the form  such that  where  is a monic and square-free polynomial in the variables . In other words, Gcd  for all  and  for all . The coefficient  is an element of the coefficients field of  and it is free of . The 's are collected with respect to .

Algebraic functions and algebraic numbers may be represented by radicals or with the RootOf notation (see type,algnum, type,algfun, type,radnum, type,radfun).

The argument  is considered as a polynomial or a rational function in . Other names are considered as elements of the coefficient field. If the optional argument  is omitted, then all the names in  which do not appear inside a RootOf or a radical are used.

The 's are positive integers if  is a polynomial in  and integers if  is a rational function in . By default, the 's may not be distinct since partial factorizations are preserved. If the option expanded is specified, factorizations of polynomials with the same multiplicity are not preserved. In any case, the 's are sorted by nondecreasing value.

Algebraic numbers and functions occurring in the results are reduced modulo their minimal polynomial (see Normal).

The RootOf and the radicals defining the algebraic numbers must form an independent set of algebraic quantities,  otherwise an error is returned. Note that this condition need not be satisfied if the expression contains only algebraic numbers in radical notation (e.g., , , ). A basis over Q for the radicals can be computed by Maple in this case.

To skip the independence checking, use the option independent.

If  or  contains functions, their arguments are normalized recursively and the functions are frozen before the computation proceeds.

Since the ordering of objects may vary from a session to another, the leading coefficients may change accordingly.

Other objects are frozen and considered as variables.

Error, (in evala/Sqrfree/preproc) reducible RootOf detected. Substitutions are {RootOf(_Z^2-_Z) = 0, RootOf(_Z^2-_Z) = 1}