  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  [33X[0;0YThis  package  provides  functionality  to  deal with one of the fundamental
  problems  in  algebra. The roots of a rational polynomial shall be expressed
  by radicals. This means one is only allowed to use the four basic operations
  [22X(+, -, ⋅ ,÷)[122X and to extract roots. For example, a radical expression for the
  roots of the polynomial [22Xx^4 - x^3 - x^2 + x + 1[122X is[133X
  
  
  [24X[33X[0;6Y\matrix{ \frac{1}{4} + \frac{1}{4}\sqrt{-3} + \frac{1}{2}\sqrt{\frac{7}{2} +
  \frac{1}{2}\sqrt{-3}}. }[133X
  
  [124X
  
  [33X[0;0YThere  are  formulas  to  solve  the general equation [22Xx^n+ a_n-1x^n-1+ ... +
  a_1x+a_0  =  0[122X up to degree 4. For higher degrees such formulae do not exist
  ([Abe26]).  It  was Évariste Galois (1811 -- 1832) who discovered that there
  exists a radical expression for the roots if and only if the Galois group of
  the  polynomial  -  initially a permutation group on the roots - is solvable
  [Gal97].  But  the  task itself was impractical in his days. This package is
  the  first  public  tool  which  provides  a  practical method for solving a
  polynomial  algebraically.  The implementation is based on Galois' ideas and
  the algorithm is described in [Dis05].[133X
  
  [33X[0;0YThe  package  can  provide  the  result  in  various  forms. As a default an
  expression is given in a similar way as in the example above. Alternatively,
  a  file  containing  the  roots  might be created which is readable by Maple
  [MGH+05].  In  [5XGAP[105X itself some information deduced during the computation is
  available.[133X
  
  [33X[0;0YThe  user  should be aware that radical expressions can get very complicated
  even  for polynomials of small degree. Especially because the algorithm will
  find  an  irreducible  radical expression. That means one gets a root of the
  given  polynomial  for  every  choice  of  a  value  of  the radicals in the
  expression.  Moreover  it  is not the aim of this package to give a simplest
  expression, in any sense.[133X
  
  [33X[0;0YIn Chapter 2 the methods provided by this package are listed and explained.[133X
  
  [33X[0;0YChapter  3  gives  details about the info class of this package. See Section
  [14X'Reference:  Info  Functions'[114X  in  the  [5XGAP[105X  reference  manual  for  general
  information about info classes.[133X
  
  [33X[0;0YWhile the installation of the package follows standard [5XGAP[105X rules the Chapter
  4  contains  information about external programs required by [5XRadiroot[105X in its
  default setup.[133X
  
  [33X[0;0YThis  package  uses  the  interface  in  the  package  [5XAlnuth[105X,  to factorise
  polynomials  over  algebraic  number  fields.  This  functionality  must  be
  available to use the functions in [5XRadiroot[105X.[133X
  
  
  [1X1.1 [33X[0;0YLicense[133X[101X
  
  [33X[0;0YThis  package  is  free  software;  you can redistribute it and/or modify it
  under  the  terms of the GNU General Public License as published by the Free
  Software  Foundation;  either  version 2 of the License, or (at your option)
  any later version.[133X
  
  [33X[0;0YThis  program is distributed in the hope that it will be useful, but WITHOUT
  ANY  WARRANTY;  without  even  the  implied  warranty  of MERCHANTABILITY or
  FITNESS  FOR  A  PARTICULAR  PURPOSE. See the GNU General Public License for
  more details.[133X
  
