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|
#############################################################################
####
##
#W Radicals.gi RADIROOT package Andreas Distler
##
## Installation file for the main function of the RADIROOT package
##
#Y 2006
##
#############################################################################
##
#F RR_RootOfUnity( <erw>, <ord> )
##
## Computes a <ord>-th root of unity built up on the roots of unity
## that already exists in the field of the record <erw>
##
InstallGlobalFunction( RR_RootOfUnity, function( erw, ord )
local i, unity, cond, faktor, m;
Info( InfoRadiroot, 2, " Finding root of unity" );
unity := One( erw.K );
if ord = 1 then
return unity;
fi;
cond := 1;
m := 1;
for i in DuplicateFreeList( Factors( ord ) ) do
# first factor of the i-th cyclotomic polynomial in H
faktor:=FactorsPolynomialAlgExt(erw.H,
CyclotomicPolynomial(Rationals,i))[1];
Info( InfoRadiroot, 3," Cyclotomic polynomial factor: ",
faktor );
if Degree( faktor ) = i-1 then
cond := cond * i; #unity := unity * E( i );
Info( InfoRadiroot, 4, " Adjoining ", i,
"-th root of unity" );
elif Degree( faktor ) = 1 then
unity := unity * Image( IsomorphismMatrixField( erw.H ),
-Value( faktor, 0 ) );
Info( InfoRadiroot, 4, " Calculate ", i,
"-th root of unity" );
else
m := i * m;
fi;
od;
if m = 1 then
return E( cond ) * unity;
else
return m * Order( unity );
fi;
end );
#############################################################################
##
#M IsSolvablePolynomial( <f> )
#M IsSolvable( <f> )
##
## Determines whether the rational polynomial <f> is solvable, e. g. whether
## its Galois group is solvable
##
InstallMethod( IsSolvablePolynomial, "for a rational polynomial",
[ IsUnivariateRationalFunction and IsPolynomial ], 0,
function( f )
if not ForAll( CoefficientsOfUnivariatePolynomial( f ), IsRat ) then
TryNextMethod( );
fi;
f := RR_SimplifiedPolynomial( f );
return ForAll( Filtered( List( Factors(f), RR_SimplifiedPolynomial ),
ff -> Degree(ff) <> 1 ),
ff -> IsSolvableGroup( TransitiveGroup( Degree(ff),
GaloisType(ff) ) ) );
end );
InstallMethod( IsSolvable, "rational polynomials", [ IsPolynomial ],
IsSolvablePolynomial );
#############################################################################
##
#M IsSeparablePolynomial( <f> )
##
## Determines whether the rational polynomial <f> is separable, e.g. whether
## it has single roots only
##
InstallMethod( IsSeparablePolynomial, "for rational polynomial",
[ IsUnivariateRationalFunction and IsPolynomial ], 0,
function( f )
if not ForAll( CoefficientsOfUnivariatePolynomial( f ), IsRat ) then
TryNextMethod( );
fi;
return Degree(Gcd( f, Derivative( f ))) = 0;
end );
#############################################################################
##
#M RootsAsMatrices( <f> )
##
## return a list of matrices with minimal polynomial <f>. The field
## generated by the matrices is a splitting field of <f>. The dimension of
## the matrices is equal to the dimension of the splitting field over the
## Rationals
##
InstallMethod( RootsAsMatrices, "rational polynomials",
[ IsUnivariateRationalFunction and IsPolynomial ], function( f )
local L, roots, erw;
if not ForAll( CoefficientsOfUnivariatePolynomial( f ), IsRat ) then
TryNextMethod( );
fi;
if not IsSeparablePolynomial( f ) then
Info(InfoWarning, 1,
"polynomial is not separable, list contains every root only once");
# make polynomial separable
f := f / Gcd( f, Derivative( f ) );
fi;
if HasSplittingField( f ) then
L := IsomorphicMatrixField( SplittingField( f ));
roots := Filtered(Basis(L), mat -> Value(f,mat) = 0*One(L));
if Length( roots ) < Degree( f ) - 1 then
erw := rec( H := SplittingField( f ), K := L );
roots:=RR_Roots([[],roots,
List(FactorsPolynomialAlgExt(SplittingField(f),f),
faktor -> -Value( faktor, 0 ) )],
erw);;
fi;
else
erw := RR_Zerfaellungskoerper(f, rec( roots := [ ],
degs := [ ],
coeffs := [ ],
K:=FieldByMatrices([ [[ 1 ]] ]),
H:=Rationals ));;
L := erw.K;
roots := RR_Roots( [ [ ], erw.roots[1], erw.roots[2] ], erw );;
fi;
Add( roots,
-CoefficientsOfUnivariatePolynomial( f )[Degree( f )]*One(L)
-Sum( roots ) );
return roots;
end );
#############################################################################
##
#F RR_Roots( <roots>, <erw> )
##
## The elements in the list of lists <roots> are in various forms. They are
## transfered in a matrix representation and returned as duplicate free list
##
InstallGlobalFunction( RR_Roots, function( roots, erw )
local i, root, B;
# Test whether there are already enough roots as matrices
if Length(roots[1]) + Length(roots[2]) >= Length(roots[3]) then
return roots[2];
fi;
B := EquationOrderBasis( erw.K, PrimitiveElement( erw.K ));
# kick out known symbolic roots
for root in Concatenation(roots[1], roots[2]) do
root := LinearCombination( Basis( erw.H ), Coefficients( B, root ) );
roots[3] := Difference( roots[3], [ root ] );
od;
# compute the other roots
if roots[1] = [ ] then Unbind(roots[3][Length(roots[3])]); fi;
for root in roots[3] do
Info(InfoRadiroot,3," Constructing ",Length(roots[2]),". root");
Add( roots[2], LinearCombination( B, ExtRepOfObj( root )));
od;
return roots[2];
end );
#############################################################################
##
#F RR_SimplifiedPolynomial( <f> )
##
## returns the polynomial g(x) with g(x^n) = f(x-a) with greatest possible n
## for the polynomial <f>
##
InstallGlobalFunction( RR_SimplifiedPolynomial, function( f )
local deg, coeff, gcd, poly;
deg := Degree( f );
poly := f / LeadingCoefficient( f );
poly := Value( poly, UnivariatePolynomial( Rationals,
[-CoefficientsOfUnivariatePolynomial(poly)[deg] / deg, 1] ) );
coeff := CoefficientsOfUnivariatePolynomial( poly );
gcd := Gcd(Filtered( [0..Degree(f)], i -> not coeff[i+1] = 0));
if gcd = 1 then
return f / LeadingCoefficient(f);
fi;
return UnivariatePolynomial(Rationals,
List([0..deg/gcd], i -> coeff[i*gcd+1]));
end );
#############################################################################
##
#F RootsOfPolynomialAsRadicals( <f>, [ <mode> , <file> ] )
#F RootsOfPolynomialAsRadicalsNC( <f>, [ <mode> , <file> ] )
##
## For the irreducible, rational polynomial <f> a representation of the
## roots as radicals is computed if this is possible, e. g. if the
## Galois group of <f> is solvable.
##
InstallGlobalFunction( RootsOfPolynomialAsRadicals, function( arg )
local f;
f := arg[1];
if Length( arg ) >= 2 and arg[2] = "off" then
if not IsSeparablePolynomial( f ) then
Error( "f must be separable" );
fi;
CallFuncList( RootsOfPolynomialAsRadicalsNC, arg );
else
# irreducibility test
if not IsIrreducible( f ) then
Error( "f must be irreducible" );
fi;
# solvibility test
if not IsSolvable( f ) then
Info( InfoRadiroot, 1, "Polynomial is not solvable." );
Info( InfoRadiroot, 3, " GaloisType is ", GaloisType( f ) );
return fail;
fi;
Info( InfoRadiroot, 3, " GaloisType is ", GaloisType( f ) );
Info( InfoRadiroot, 2, " Galoisgroup is ",
TransitiveGroup( Degree( f ), GaloisType( f )));
return CallFuncList( RootsOfPolynomialAsRadicalsNC, arg );
fi;
end );
InstallGlobalFunction( RootsOfPolynomialAsRadicalsNC, function( arg )
local erw,elements,lcm,conj,bas,file,dir,poly,B,fix,compser,f,mode,path;
f := arg[1];
if 1 = Length( arg ) then
mode := "dvi";
else
mode := arg[2];
fi;
while not mode in [ "off", "dvi", "maple", "latex" ] do
Error( "<mode> has to be a valid option" );
od;
# normed, simplified polynomial
if mode <> "off" then
# irreducibility test
if not IsIrreducible( f ) then
Error( "f must be irreducible" );
fi;
poly := RR_SimplifiedPolynomial( f );
Info( InfoRadiroot, 2, " Normed, simplified Polynomial: ", poly );
else
if LeadingCoefficient( f ) <> 1 then
Error( "f must be a normed polynomial" );
fi;
poly := f;
fi;
Info( InfoRadiroot, 2, " Construction of the splitting field" );
erw := RR_Zerfaellungskoerper( poly, rec( roots := [ ],
degs := [ ],
coeffs := [ ],
K:=FieldByMatrices([ [[ 1 ]] ]),
H:=Rationals ));;
# get all roots, set a basis of the primitive element
erw.roots := RR_Roots( [ [], erw.roots[1], erw.roots[2] ], erw );;
Add( erw.roots,
-CoefficientsOfUnivariatePolynomial(poly)[Degree(poly)]*One(erw.K)
-Sum( erw.roots ) );
SetRootsAsMatrices( poly, erw.roots );
# get structure of primitive element to use RR_Produkt
erw.coeffs := Filtered(Coefficients(Basis(erw.K),PrimitiveElement(erw.K)),
i -> i <> 0 );
# for mode "off" it remains to compute the Galois group
if mode = "off" then
if not HasGaloisGroupOnRoots( poly ) then
erw.unity := 1;
erw.galgrp := RR_ConstructGaloisGroup( erw );
SetGaloisGroupOnRoots( poly, erw.galgrp );
fi;
return;
fi;
# try to find root of unity, if fail start all over
erw.unity := RR_RootOfUnity( erw, DegreeOverPrimeField(erw.K) );
if IsInt(erw.unity) then
erw := RR_SplittField(poly, erw.unity );
# need roots in bigger field
erw.roots := RR_Roots( [ [], erw.roots[1], erw.roots[2] ], erw );;
Add(erw.roots,
-CoefficientsOfUnivariatePolynomial(poly)[Degree(poly)]
*One(erw.K) - Sum( erw.roots ) );
erw.coeffs := Filtered( Coefficients( Basis( erw.K ),
PrimitiveElement( erw.K )),
i -> i <> 0 );
erw.galgrp := RR_ConstructGaloisGroup( erw );
elif HasGaloisGroupOnRoots( poly ) then
erw.galgrp := GaloisGroupOnRoots( poly );
else
erw.galgrp := RR_ConstructGaloisGroup( erw );
SetGaloisGroupOnRoots( poly, erw.galgrp );
fi;
Info( InfoRadiroot, 2, " Galoisgroup as PermGrp is ", erw.galgrp );
if not IsSolvable( erw.galgrp ) then
Info( InfoRadiroot, 1, "Polynomial is not solvable." );
return fail;
fi;
Info(InfoRadiroot,4," h := Lcm( Order( Galoisgroup ) ) = ",
Product(Unique(Factors(Order(erw.galgrp)))) );
if IsDiagonalMat( erw.unity ) then
Info( InfoRadiroot, 3,
" no root of unity in the splitting field");
compser := CompositionSeries( erw.galgrp );
elif Length( erw.degs ) <> Length( erw.coeffs ) then
compser := CompositionSeries( erw.galgrp );
else
fix := Filtered(AsList(erw.galgrp),
p -> RR_Produkt(erw, erw.unity, p) = erw.unity);
compser := RR_CompositionSeries( erw.galgrp, AsGroup( fix ));
fi;
erw.K!.cyclics := RR_CyclicElements( erw, compser );;
Info( InfoRadiroot, 2, " computed cyclic elements" );
if 3 = Length( arg ) then
file := arg[3];
dir := DirectoryCurrent( );
else
dir := DirectoryTemporary( );
file := "Nst";
fi;
if mode <> "maple" then
if 3 = Length(arg) and IsExistingFile(Concatenation(file, ".tex")) then
Error( file, ".tex already exists" );
fi;
path := RR_TexFile( f, erw, erw.K!.cyclics, dir,
Concatenation( file, ".tex" ) );
if mode = "dvi" then
RR_Display( file, dir );
fi;
else
if 3 = Length(arg) and IsExistingFile(file) then
Error( file, " already exists" );
fi;
path := RR_MapleFile( f, erw, erw.K!.cyclics, Filename(dir,file));
fi;
return path;
end );
#############################################################################
##
#E
|
