|
#########################################################################
#
# this file contains the methods for defining the field SqrtField
# and for working with its elements, see sqrt.gd
#
#
###############################################
# Do deal with Flat for lists of matrices
#
corelg.myflat := function(L)
return Concatenation(L);
end;
###############################################
# DUE TO A BUG IN BasisNC
InstallMethod( NiceBasisNC,
[IsBasisByNiceBasis and HasNiceBasis], NiceBasis );
#########################################################################
BindGlobal( "SqrtField_objectify", function( elm )
local u,isrt;
isrt:= Length(elm) = 1 and Length(elm[1][2])=0;
u:= Objectify( SqrtFieldType, [ Immutable(elm) ] );
u![2]:= isrt;
return u;
end );
#########################################################################
InstallMethod( \in,
"for an object and a SqrtField",
[ IsObject, IsSqrtField ], 1000000,
function( p, qf )
return IsSqrtFieldElement(p);
end );
#########################################################################
InstallGlobalFunction(Sqroot, function(q)
local d, n, fc, cf, ps, i, m, sgn;
if not IsRat(q) then
if IsSqrtFieldElement(q) and Length(q![1]) = 1
and IsRat(q![1][1][1]) and q![1][1][2] = [] then
return Sqroot(q![1][1][1]);
else
Error("<q> must be a rational, or rational SqrtField elt");
fi;
fi;
if q < 0 then sgn := E(4); q := -q; else sgn := 1; fi;
if q in [0,1] then
return SqrtField_objectify([[sgn*q,[]]]);
fi;
d := DenominatorRat(q);
n := NumeratorRat(q);
fc := FactorsInt(n);
Append(fc,FactorsInt(d));
fc := Collected(fc);
cf := 1/d;
ps := [];
for i in [1..Length(fc)] do
m := fc[i][2] mod 2;
cf := cf *(fc[i][1]^((fc[i][2]-m)/2));
if m=1 then Add(ps,fc[i][1]); fi;
od;
if ps = [] then return
SqrtField_objectify([[sgn*cf,[]]]);
fi;
if ps[1] = 1 then ps := ps{[2..Length(ps)]}; fi;
Sort(ps);
MakeImmutable(ps);
return SqrtField_objectify([[sgn*cf,ps]] );
end);
#########################################################################
SetOne( SqrtField, Sqroot(1) );
SetZero( SqrtField, Sqroot(0) );
SetZero( SqrtFieldFam, Zero(SqrtField) );
SetOne( SqrtFieldFam, One(SqrtField) );
SetIsUFDFamily(SqrtFieldFam,true);
#########################################################################
InstallMethod( ViewObj,
"for SqrtField elements",
[ IsSqrtFieldElement ],
function( sf )
local elt, l, i, printmon, printcf;
elt := sf![1];
l := Length(elt);
printmon := function(mon)
if mon[1] = 1 then
Print( "Sqroot(",Product(mon[2]),")");
elif mon[1]<0 or not (IsRat(mon[1]) or IsRat(E(4)*mon[1]))
or (IsRat(E(4)*mon[1]) and E(4)*mon[1]>0) then
Print( "(",mon[1],")*","Sqroot(",Product(mon[2]),")");
else
Print( mon[1],"*","Sqroot(",Product(mon[2]),")");
fi;
end;
if l=1 then
if elt[1][2]=[] then Print(elt[1][1]); else printmon(elt[1]); fi;
else
if elt[1][2]=[] then
Print(elt[1][1]," + ");
else
printmon(elt[1]);
Print(" + ");
fi;
fi;
for i in [2..l-1] do printmon(elt[i]); Print(" + "); od;
if l>1 then printmon(elt[l]); fi;
end );
##########################################################################
InstallMethod( PrintObj,
"for SqrtField elements",
[ IsSqrtFieldElement ],
function( sf )
local elt, l, i, printmon;
elt := sf![1];
l := Length(elt);
printmon := function(mon)
if mon[1] = 1 then
Print( "Sqroot(",Product(mon[2]),")");
elif mon[1]<0 or not (IsRat(mon[1]) or IsRat(E(4)*mon[1]))
or (IsRat(E(4)*mon[1]) and E(4)*mon[1]>0) then
Print( "(",mon[1],")*","Sqroot(",Product(mon[2]),")");
else
Print( mon[1],"*","Sqroot(",Product(mon[2]),")");
fi;
end;
if l=1 then
if elt[1][2]=[] then Print(elt[1][1]); else printmon(elt[1]); fi;
else
if elt[1][2]=[] then
Print(elt[1][1]," + ");
else
printmon(elt[1]);
Print(" + ");
fi;
fi;
for i in [2..l-1] do printmon(elt[i]); Print(" + "); od;
if l>1 then printmon(elt[l]); fi;
end );
#########################################################################
InstallGlobalFunction( SqrtFieldIsGaussRat, function(el)
return IsSqrtFieldElement(el) and el![2];
end);
#########################################################################
InstallGlobalFunction( SqrtFieldMakeRational, function(el)
if IsSqrtFieldElement(el) then
if SqrtFieldIsGaussRat(el) then
return el![1][1][1];
else
return false;
fi;
elif IsSqrtFieldElementCollection(el) then
if ForAll(el,SqrtFieldIsGaussRat) then
return List(el,x->x![1][1][1]);
else
return false;
fi;
elif IsSqrtFieldElementCollColl(el) then
if ForAll(Flat(el),SqrtFieldIsGaussRat) then
return List(el,x->List(x,y->y![1][1][1]));
else
return false;
fi;
fi;
return false;
end);
#########################################################################
InstallMethod( Determinant,
"for SqrtField matrices",
[ IsSqrtFieldElementCollColl], 1000,
function(x)
local a,b;
a := SqrtFieldMakeRational(x);
if not a = false then
return DeterminantMat(a)*One(SqrtField);
fi;
TryNextMethod();
end);
#########################################################################
InstallMethod( \*,
"for SqrtField matrices",
[ IsSqrtFieldElementCollColl, IsSqrtFieldElementCollColl],
function(x,y)
local a,b;
a := SqrtFieldMakeRational(x);
if not a = false then
b := SqrtFieldMakeRational(y);
if not b = false then
return One(SqrtField)*(a*b);
fi;
fi;
#Print("DONT USE MAKERAT\n");
TryNextMethod();
end);
#########################################################################
InstallMethod( \+,
"for SqrtField matrices",
[ IsSqrtFieldElementCollColl, IsSqrtFieldElementCollColl],
function(x,y)
local a,b;
a := SqrtFieldMakeRational(x);
if not a = false then
b := SqrtFieldMakeRational(y);
if not b = false then
return One(SqrtField)*(a+b);
fi;
fi;
#Print("DONT USE MAKERAT\n");
TryNextMethod();
end);
#########################################################################
InstallMethod( \*,
"for rational and SqrtField elements",
[IsCyclotomic, IsSqrtFieldElement],
function(x,y)
local b,i;
if not IsGaussRat(x) then
Error("cyclotomic has to be Gaussian Rational");
fi;
if x = 0 then return Zero(SqrtField); fi;
if x = 1 then return y; fi;
b := List(y![1],ShallowCopy);
for i in b do i[1] := x*i[1]; od;
return SqrtField_objectify( b );
end);
#########################################################################
InstallMethod( \*,
"for SqrtField and rational elements",
[IsSqrtFieldElement, IsCyclotomic],
function(x,y)
return y*x;
end);
#########################################################################
InstallMethod( ZeroOp,
"for SqrtField element",
[IsSqrtFieldElement],
function(x)
return Zero(SqrtField);
end);
#########################################################################
InstallMethod( OneOp,
"for SqrtField element",
[IsSqrtFieldElement],
function(x)
return One(SqrtField);
end);
#########################################################################
InstallMethod( \+,
"for SqrtField and SqrtField elements",
[IsSqrtFieldElement, IsSqrtFieldElement], 10000,
function(x,y)
local a,b,i,j,e,len,pr;
if x = Zero(SqrtField) then return y; fi;
if y = Zero(SqrtField) then return x; fi;
if x![2] then
if y![2] then
return SqrtField_objectify([[x![1][1][1]+y![1][1][1], []]]);
else
return x![1][1][1] + y;
fi;
fi;
if y![2] then return y![1][1][1] + x; fi;
if Length(x![1])<=Length(y![1]) then
a := List(x![1],ShallowCopy);
b := List(y![1],ShallowCopy);
else
a := List(y![1],ShallowCopy);
b := List(x![1],ShallowCopy);
fi;
i := 1;
for j in [1..Length(a)] do
e := a[j];
pr := Product(e[2]);
while i<= Length(b) and Product(b[i][2])<pr do i := i+1; od;
if i>Length(b) then
Append(b,a{[j..Length(a)]});
return SqrtField_objectify( b );
fi;
if e[2]=b[i][2] then
b[i][1] := b[i][1] + e[1];
if b[i][1] = 0 then
#remove position i
len := Length(b);
CopyListEntries( b, i + 1, 1, b, i, 1, len - i );
Unbind( b[len] );
fi;
else
#add e at position i
CopyListEntries(b,i,1,b,i+1,1,Length(b)-i+1);
b[i] := e;
fi;
od;
if b = [] then return Zero(SqrtField); fi;
return SqrtField_objectify( b );
end);
#########################################################################
InstallMethod( \+,
"for Gaussian rational and SqrtField elements",
[IsCyclotomic, IsSqrtFieldElement],
function(x,y)
local t;
if not IsGaussRat(x) then
Error("cyclotomic has to be Gaussian rational");
fi;
if x=0 then return y; fi;
t := List(y![1],ShallowCopy);
if t[1][2] = [] then
t[1][1] := t[1][1] + x;
if t[1][1] = 0 then t := t{[2..Length(t)]}; fi;
if t = [] then return Zero(SqrtField); fi;
else
#add [y,[]] at position 1
CopyListEntries( t, 1, 1, t, 2, 1, Length(t) );
t[1] := [x,[]];
fi;
return SqrtField_objectify( t );
end);
#########################################################################
InstallMethod( \+,
"for SqrtField and rational elements",
[IsSqrtFieldElement, IsCyclotomic],
function(x,y)
return y+x;
end);
#########################################################################
InstallMethod( AdditiveInverseSameMutability,
"for SqrtField elements",
[IsSqrtFieldElement],
function(x)
local a, i;
a := List(x![1],ShallowCopy);
for i in a do i[1] := -i[1]; od;
return SqrtField_objectify( a );
end);
#########################################################################
InstallMethod( \=,
"for SqrtField elements",
[IsSqrtFieldElement, IsSqrtFieldElement],
function(x,y)
return x![1] = y![1];
end);
#########################################################################
InstallMethod( \=,
"for SqrtField and rational elements",
[IsSqrtFieldElement, IsRat],
function(x,y)
if not x![2] then return false; fi;
return SqrtFieldMakeRational(x) = y;
end);
#########################################################################
InstallMethod( \=,
"for rational and SqrtField element",
[IsRat, IsSqrtFieldElement],
function(x,y)
if not y![2] then return false; fi;
return SqrtFieldMakeRational(y) = x;
end);
#########################################################################
InstallMethod( \*,
"for SqrtField and SqrtField elements",
[IsSqrtFieldElement, IsSqrtFieldElement],
function(x,y)
local a,b,i,j,res,int,cf,v,w, mns, cfs, isSmaller, pos, k;
if x=Zero(SqrtField) or y=Zero(SqrtField) then
return Zero(SqrtField);
fi;
if x=One(SqrtField) then return y; fi;
if y=One(SqrtField) then return x; fi;
a := x![1];
b := y![1];
if x![2] and y![2] then
return SqrtField_objectify([[a[1][1]*b[1][1],[]]]);
fi;
mns := [ ];
cfs := [ ];
isSmaller := function(u,v) return Product(u)< Product(v); end;
for i in a do
for j in b do
int := i[2]{[1..Length(i[2])]};
INTER_SET( int, j[2] );
cf := i[1]*j[1]; #*Product(int);
for k in int do cf:= cf*k; od;
res := i[2]{[1..Length(i[2])]};
APPEND_LIST( res, j[2] );
v := Filtered(res,x->not x in int);
SORT_LIST(v);
pos := POSITION_SORTED_LIST_COMP( mns, v, isSmaller );
if IsBound(mns[pos] ) and mns[pos] = v then
cfs[pos]:= cfs[pos]+cf;
else
Add( mns, v, pos );
Add( cfs, cf, pos );
fi;
od;
od;
res:= [ ];
for i in [1..Length(mns)] do
if cfs[i] <> 0 then
Add( res, [ cfs[i], mns[i] ] );
fi;
od;
if res = [] then return Zero(SqrtField); fi;
return SqrtField_objectify( res );
end);
##########################################################################
InstallMethod( InverseOp,
"for SqrtField elements",
[IsSqrtFieldElement],
function(el)
local b, i, j, bas, mat, elts, a, cfs, l,cf, pos, ns, inv, res, v, nrp, want;
if el=Zero(SqrtField) then Error("elt must be nonzero"); fi;
a := List(el![1],ShallowCopy);
a := SqrtField_objectify( a );
i := 1;
elts := [One(SqrtField)];
bas := [[]];
cfs := [ [1] ];
l := Length(bas);
nrp := Length(Collected(Flat(List(a![1],x->x[2]))));
want := 2^Minimum([nrp,Length(Filtered(a![1],x->not x[2]=[]))]);
if want > 16 then
Info(InfoSqrtField,2,"need ",want," powers for computing the inverse");
fi;
repeat
i := i+1;
elts[i] := elts[i-1]*a;
cfs[i] := ListWithIdenticalEntries(l,0);
for j in elts[i]![1] do
pos := Position(bas,j[2]);
if pos=fail then
Add(bas,j[2]);
l := l+1;
pos := l;
for v in cfs do v[l] := 0; od;
fi;
cfs[i][pos] := j[1];
od;
if i mod 10 = 0 then
Info(InfoSqrtField,2," InfoSqrtField: (inverses) computed ",i," powers");
fi;
until i-1 = want;
ns := NullspaceMat(cfs);
res := ns[1];
inv := Sum(List([2..Length(res)],i->res[i]*elts[i-1]))/(-res[1]);
#Info(InfoSqrtField,3,"test inverse");
#if not el*inv = One(SqrtField) then Error("hmpf... inv wrong!"); fi;
return inv;
end);
#########################################################################
InstallMethod( AdditiveInverseMutable,
"for SqrtField elements",
[IsSqrtFieldElement],
function(x)
local a, i;
a := List(x![1],ShallowCopy);
for i in a do i[1] := -i[1]; od;
return SqrtField_objectify( a );
end);
#########################################################################
InstallMethod( \<,
"for SqrtField elements",
[IsSqrtFieldElement, IsSqrtFieldElement],
function(x,y)
return x![1] < y![1];
end);
#########################################################################
InstallMethod( \<,
"for SqrtField element and Rational",
[IsSqrtFieldElement, IsRat],
function(x,y)
if not x![2] then TryNextMethod(); fi;
return SqrtFieldMakeRational(x) < y;
end);
########################################################################
InstallMethod( \<,
"for rational and SqrtField element",
[IsRat, IsSqrtFieldElement],
function(x,y)
if not y![2] then TryNextMethod(); fi;
return SqrtFieldMakeRational(y) > x;
end);
#########################################################################
InstallGlobalFunction(SqrtFieldMinimalPolynomial, function(el)
#returns polynomial f of min deg over GaussianRats such that f(el)=0;
local b, i, j, bas, mat, elts, a, cfs, l,cf, pos, ns, inv, res, v, nrp,
want, pol, x;
if not IsSqrtFieldElement(el) then
Error("input has to be SqrtField elt");
fi;
if el=Zero(SqrtField) then Error("elt must be nonzero"); fi;
a := List(el![1],ShallowCopy);
a := SqrtField_objectify( a );
i := 1;
elts := [One(SqrtField)];
bas := [[]];
cfs := [ [1] ];
l := Length(bas);
nrp := Length(Collected(Flat(List(a![1],x->x[2]))));
want := 2^Minimum([nrp,Length(Filtered(a![1],x->not x[2]=[]))]);
if want > 16 then
Info(InfoSqrtField,1,"need ",want," powers for computing the inverse");
fi;
repeat
i := i+1;
elts[i] := elts[i-1]*a;
cfs[i] := ListWithIdenticalEntries(l,0);
for j in elts[i]![1] do
pos := Position(bas,j[2]);
if pos=fail then
Add(bas,j[2]);
l := l+1;
pos := l;
for v in cfs do v[l] := 0; od;
fi;
cfs[i][pos] := j[1];
od;
if i mod 10 = 0 then
Info(InfoSqrtField,2," - computed ",i," elements");
fi;
until i-1 = want;
ns := NullspaceMat(cfs);
res := ns[1]*One(SqrtField);
x := Indeterminate(SqrtField);
elts := [One(SqrtField)];
Append(elts, List([2..Length(res)],i->x^(i-1)));
pol := Sum(List([1..Length(res)],i->res[i]*elts[i]));
return pol;
end);
#########################################################################
InstallGlobalFunction(SqrtFieldEltByRationalSqrt, function(e)
local sgn;
if not IsRat(e^2) then Error("input has to be Sqroot of a rational"); fi;
if not e-Sqrt(e^2) = 0 then sgn := -1; else sgn := 1; fi;
return sgn*Sqroot(e^2);
end);
##########################################################################
InstallGlobalFunction(SqrtFieldEltRealAndComplexPart, function(e)
local real, imag, i, makecf, cf, r;
if not IsSqrtFieldElement(e) then
Error("input must be SqrtFieldElement");
fi;
makecf := function(cf)
local cj;
cj := ComplexConjugate(cf);
return [1/2*(cf+cj),1/2*(cf-cj)];
end;
real := Zero(SqrtField);
imag := Zero(SqrtField);
for i in e![1] do
r := Sqroot(Product(i[2]));
cf := makecf(i[1]);
real := real+cf[1]*r;
imag := imag+cf[2]*r;
od;
return [real,imag];
end);
##########################################################################
InstallGlobalFunction(IsPosSqrtFieldElt, function(e)
local real, imag, i, makecf, cf, r;
if not IsSqrtFieldElement(e) or not Length(e![1]) = 1 or
not ForAll(e![1],i->IsRat(i[1])) then
Error("input must be a real SqrtFieldElement with one monom");
fi;
return e![1][1][1]>0;
end);
##########################################################################
InstallGlobalFunction(CoefficientsOfSqrtFieldElt, function(e)
local cfs, i, x;
if not IsSqrtFieldElement(e) then
Error("input must be SqrtFieldElement");
fi;
x := List(e![1],ShallowCopy);
return List(x,i->[i[1],Product(i[2])]);
end);
##########################################################################
InstallGlobalFunction(SqrtFieldEltByCoefficients, function(e)
local elt, i;
elt := Zero(SqrtField);
for i in e do
if not IsGaussRat(i[1]) and IsRat(i[2]) then
Error("wrong format of coefficients vector");
fi;
elt := elt + i[1]*Sqroot(i[2]);
od;
return elt;
end);
##########################################################################
InstallGlobalFunction(SqrtFieldEltToCyclotomic, function(e)
local makemon;
if not IsSqrtFieldElement(e) then
Error("input has to lie in SqrtField");
fi;
if Length(e![1])=1 and e![1][1][2]=[] then return e![1][1][1]; fi;
#Info(InfoSqrtField,1,"Warning: might not work for large cyclotomics.");
makemon := function(mon)
if mon[2]=[] then return mon[1]; fi;
return mon[1]*Sqrt(Product(mon[2]));
end;
return Sum(List(e![1],makemon));
end);
##########################################################################
InstallMethod( DefaultFieldOfMatrix,
"method for a matrix over SqrtField",
[ IsMatrix and IsSqrtFieldElementCollColl ],
function( mat )
return SqrtField;
end );
##########################################################################
InstallMethod( DefaultFieldByGenerators,
"method for a list over SqrtField",
[ IsList and IsSqrtFieldElementCollection ],
function( mat )
return SqrtField;
end );
##########################################################################
InstallMethod( ComplexConjugate,
"method for SqrtField elt",
[ IsSqrtFieldElement ],
function( el )
local x, i;
x := List(el![1], ShallowCopy);
for i in x do i[1] := ComplexConjugate(i[1]); od;
return SqrtField_objectify( x );
end );
##########################################################################
InstallMethod( Sqrt,
"method for SqrtField elt",
[ IsSqrtFieldElement ],
function( el )
local x, i;
x := List(el![1],ShallowCopy);
if not (Length(x) = 1 and x[1][2] = [] and IsRat(x[1][1])) then
Error("input has to be rational SqrtFieldElt");
fi;
return Sqroot(x[1][1]);
end );
##########################################################################
InstallMethodWithRandomSource( Random,
"for a random source and an SqrtField ",
[ IsRandomSource, IsSqrtField ],
function( rs, F )
return Sum(List([1..15],x-> Random(rs,Rationals)*Sqroot(Random(rs,Rationals))));
end );
##########################################################################
InstallMethod( AbsoluteValue,
"for SqrtField element",
[ IsSqrtFieldElement ], 10000,
function( el )
local x,i;
x := List(el![1],ShallowCopy);
if not (Length(x)=1 and IsRat(x[1][1])) then
Error("elt must be a real SqrtFieldElt with one summand");
fi;
for i in x do i[1] := AbsoluteValue(i[1]); od;
return SqrtField_objectify( x );
end );
##########################################################################
InstallMethod( String,
"for SqrtField element",
[ IsSqrtFieldElement ],
function( el )
local elt, l, i, printmon, printcf, str;
elt := List(el![1],ShallowCopy);
str := "";
l := Length(elt);
printmon := function(mon)
if mon[1] = 1 then
Append(str, "Sqroot(");
Append(str,String(Product(mon[2])));
Append(str,")");
elif mon[1]<0 or not (IsRat(mon[1]) or IsRat(E(4)*mon[1]))
or (IsRat(E(4)*mon[1]) and E(4)*mon[1]>0) then
Append(str,"(");
Append(str,String(mon[1]));
Append(str,")*Sqroot(");
Append(str,String(Product(mon[2])));
Append(str,")");
else
Append(str, String(mon[1]));
Append(str,"*Sqroot(");
Append(str,String(Product(mon[2])));
Append(str,")");
fi;
end;
if l=1 then
if elt[1][2]=[] then Append(str,String(elt[1][1])); else printmon(elt[1]); fi;
else
if elt[1][2]=[] then
Append(str,String(elt[1][1]));
Append(str," + ");
else
printmon(elt[1]);
Append(str," + ");
fi;
fi;
for i in [2..l-1] do printmon(elt[i]); Append(str," + "); od;
if l>1 then printmon(elt[l]); fi;
return str;
end );
##########################################################################
InstallMethod( String,
"for SqrtField",
[ IsSqrtField ],
function( el )
return "SqrtField";
end );
##########################################################################
InstallGlobalFunction(SqrtFieldEltByCyclotomic, function(arg)
local e, cj, re, im, solvereal, res, r1, r2,
sqfree, whichSqfreeRootsInCF;
e := arg[1];
if e in SqrtField then return e; fi;
if not IsCyclotomic(e) then
Error("input must be Cyclotomic");
fi;
if IsGaussRat(e) then return e*One(SqrtField); fi;
whichSqfreeRootsInCF := function(n)
#returns also one for technical reasons
local sqfree, eeven, res;
sqfree := x -> ForAll(Collected(FactorsInt(x)),i->i[2]=1);
eeven := x -> Length( Filtered(List(Collected(FactorsInt(x)), i->i[1]),
p -> p mod 4 = 3)) mod 2 = 0;
res := Filtered(DivisorsInt(n),x->sqfree(x));
if IsOddInt(n) or (IsInt(n/2) and IsOddInt(n/2)) then
return Filtered(res,x->IsOddInt(x) and eeven(x));
fi;
if n mod 4 = 0 and not n mod 8 =0 then
return Filtered(res,IsOddInt);
fi;
return res;
end;
#check if input is sqrt of rational
#test is expensive, e.g. for e=Sqrt(1231)+Sqrt(17);;
#if IsRat(e^2) then return SqrtFieldEltByRationalSqrt(e); fi;
#split into real and complex part
cj := ComplexConjugate(e);;
re := 1/2*(e+cj);;
im := 1/2*E(4)^3*(e-cj);;
#the main function which requires a real input
solvereal := function( r )
local n, pr, prsq, bas, mat, ns;
if IsRat(r) then return r*One(SqrtField); fi;
n := Conductor(r);
pr := whichSqfreeRootsInCF(n);
prsq := List(pr,Sqrt);
bas := Basis(CF(n));
mat := List(prsq,x->Coefficients(bas,x));
ns := SolutionMat(mat,Coefficients(bas,r));
if ns = fail then return fail; fi;
return Sum(List([1..Length(pr)],i->ns[i]*Sqroot(pr[i])));
end;
#try to solve real and complex part
r1 := solvereal(re); if r1 = fail then return fail; fi;
r2 := solvereal(im); if r2 = fail then return fail; fi;
res := r1 + E(4)*r2;
###TEST
#Info(InfoSqrtField,1,"test res");
#if not e = SqrtFieldEltToCyclotomic(res) then
# Error("res wrong");
#fi;
return res;
end);
#########################################################################
InstallGlobalFunction( SqrtFieldPolynomialToRationalPolynomial, function(e)
local x,i,j,cf;
x := Indeterminate(Rationals);
cf := List(CoefficientsOfUnivariatePolynomial(e),SqrtFieldEltToCyclotomic);
return Sum(List([1..Length(cf)],i-> cf[i]*x^(i-1)));
end);
#########################################################################
InstallGlobalFunction( SqrtFieldRationalPolynomialToSqrtFieldPolynomial, function(e)
local x,i,j,cf;
x := Indeterminate(SqrtField);
cf := List(CoefficientsOfUnivariatePolynomial(e),SqrtFieldEltByCyclotomic);
return Sum(List([1..Length(cf)],i-> cf[i]*x^(i-1)));
end);
##########################################################################
InstallOtherMethod( Factors,
"for SqrtField",
[ IsPolynomialRing, IsUnivariatePolynomial ],
function( R, f )
local rf;
if LeftActingDomain(R) = SqrtField and
IsSqrtFieldElementCollection(CoefficientsOfUnivariatePolynomial(f)) then
rf := SqrtFieldPolynomialToRationalPolynomial(f);
rf := Factors(rf);
return List(rf,SqrtFieldRationalPolynomialToSqrtFieldPolynomial);
else
TryNextMethod();
fi;
end );
[ Dauer der Verarbeitung: 0.42 Sekunden
(vorverarbeitet)
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