%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%W  radiroot.tex          Radiroot documentation          Andreas Distler
%%
%H  $Id:$
%%
%Y  2005
%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Chapter{Functionality of the package}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Methods for rational polynomials}

\> IsSolvable( <f> )
\> IsSolvablePolynomial( <f> )

returns `true' if the rational polynomial <f> has a solvable Galois group and
`false' otherwise. It signals an error if there exists an irreducible factor
with degree greater than 15.

For a rational polynomial <f>
\> SplittingField( <f> )

returns the smallest field, constructed with `FieldByPolynomial', that
contains all roots of <f>. 

\> GaloisGroupOnRoots( <f> )

calculates the Galois group <G> of the rationals polynomial <f> as a
permutation group with respect to the ordering of the roots of <f> given as
matrices in <G!.roots>.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Solving a polynomial by radicals}

\> RootsOfPolynomialAsRadicals( <f> )

computes a solution by radicals for the irreducible, rational polynomial <f>
up to degree 15 if this is possible, e. g. if the Galoisgroup of <f> is
solvable. The result is displayed in form of a dvi-file. Additionally a record
is returned which contains the roots of <f> as a list `roots' of matrices, the
Galois group on the roots as component `galgrp'
and the splitting field of <f> in two forms; on the one hand the
matrix field  `K' generated by the roots and on the other hand an algebraic
number field  `H' created by the defining polynomial of the matrix field. The
record also includes a list `cyclics' of matrices which define the splitting
field by gradual, cyclic extensions.

The computation may last very long and doesn't finish for every example if the
degree of <f> is greater than 7.

\> RootsOfPolynomialAsRadicalsNC( <f>, <display> )

does essentially the same as `RootsOfPolynomialAsRadicals' except that you can
choose if you want to create a dvi-file and display it by setting the boolean
<display>. The function performs no test whether the polynomial <f> is
irreducible. It also doesn't check at the begining if <f> is solvable, but can
therefore be used for polynomials with arbitrary degree.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%E





