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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Martin Schönert.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains methods for Gaussian rationals and Gaussian integers.
##
## Gaussian rationals are elements of the form $a + b * I$ where $I$ is the
## square root of -1 and $a,b$ are rationals.
## Note that $I$ is written as `E(4)', i.e., as a fourth root of unity in
## {\GAP}.
## Gauss was the first to investigate such numbers, and already proved that
## the ring of integers of this field, i.e., the elements of the form
## $a + b * I$ where $a,b$ are integers, forms a Euclidean Ring.
## It follows that this ring is a Unique Factorization Domain.
##
#############################################################################
##
#V GaussianIntegers . . . . . . . . . . . . . . . ring of Gaussian integers
##
BindGlobal( "GaussianIntegers", Objectify( NewType(
CollectionsFamily(CyclotomicsFamily),
IsGaussianIntegers and IsAttributeStoringRep ),
rec() ) );
SetLeftActingDomain( GaussianIntegers, Integers );
SetName( GaussianIntegers, "GaussianIntegers" );
SetString( GaussianIntegers, "GaussianIntegers" );
SetIsLeftActedOnByDivisionRing( GaussianIntegers, false );
SetSize( GaussianIntegers, infinity );
SetGeneratorsOfRing( GaussianIntegers, [ E(4) ] );
SetGeneratorsOfLeftModule( GaussianIntegers, [ 1, E(4) ] );
SetIsFinitelyGeneratedMagma( GaussianIntegers, false );
SetUnits( GaussianIntegers, [ -1, 1, -E(4), E(4) ] );
SetIsWholeFamily( GaussianIntegers, false );
#############################################################################
##
#V GaussianRationals . . . . . . . . . . . . . . field of Gaussian rationals
##
BindGlobal( "GaussianRationals", Objectify( NewType(
CollectionsFamily( CyclotomicsFamily ),
IsGaussianRationals and IsAttributeStoringRep ), rec() ) );
SetName( GaussianRationals, "GaussianRationals" );
SetLeftActingDomain( GaussianRationals, Rationals );
SetIsPrimeField( GaussianRationals, false );
SetIsCyclotomicField( GaussianRationals, true );
SetSize( GaussianRationals, infinity );
SetConductor( GaussianRationals, 4 );
SetDimension( GaussianRationals, 2 );
SetDegreeOverPrimeField( GaussianRationals, 2 );
SetGaloisStabilizer( GaussianRationals, [ 1 ] );
SetGeneratorsOfLeftModule( GaussianRationals, [ 1, E(4) ] );
SetIsFinitelyGeneratedMagma( GaussianRationals, false );
SetIsWholeFamily( GaussianRationals, false );
#############################################################################
##
#M \in( <n>, GaussianIntegers ) . . . membership test for Gaussian integers
##
## Gaussian integers are of the form `<a> + <b> * E(4)', where <a> and <b>
## are integers.
##
InstallMethod( \in,
"for Gaussian integers",
IsElmsColls,
[ IsCyc, IsGaussianIntegers ],
function( cyc, GaussianIntegers )
return IsCycInt( cyc ) and 4 mod Conductor( cyc ) = 0;
end );
#############################################################################
##
#M Basis( GaussianIntegers ) . . . . . . . . . . . . . for Gaussian integers
##
InstallMethod( Basis,
"for Gaussian integers (delegate to `CanonicalBasis')",
[ IsGaussianIntegers ], CANONICAL_BASIS_FLAGS,
CanonicalBasis );
#############################################################################
##
#M CanonicalBasis( GaussianIntegers ) . . . . . . . . for Gaussian integers
##
DeclareRepresentation(
"IsCanonicalBasisGaussianIntegersRep", IsAttributeStoringRep,
[ "conductor", "zumbroichbase" ] );
InstallMethod( CanonicalBasis,
"for Gaussian integers",
[ IsGaussianIntegers ],
function( GaussianIntegers )
local B;
B:= Objectify( NewType( FamilyObj( GaussianIntegers ),
IsFiniteBasisDefault
and IsCanonicalBasis
and IsCanonicalBasisGaussianIntegersRep ),
#T generalize this to integral rings of cyclotomics!
rec() );
SetUnderlyingLeftModule( B, GaussianIntegers );
SetIsIntegralBasis( B, true );
SetBasisVectors( B, Immutable( [ 1, E(4) ] ) );
B!.conductor:= 4;
B!.zumbroichbase := [ 0, 1 ];
# Return the basis.
return B;
end );
#############################################################################
##
#M Coefficients( <B>, <z> ) . for the canon. basis of the Gaussian integers
##
InstallMethod( Coefficients,
"for canon. basis of Gaussian integers, and cyclotomic",
IsCollsElms,
[ IsBasis and IsCanonicalBasis and IsCanonicalBasisGaussianIntegersRep,
IsCyc ],
function( B, z )
local N,
coeffs,
F;
F:= UnderlyingLeftModule( B );
if not z in F then return fail; fi;
N:= B!.conductor;
# Get the Zumbroich basis representation of <z> in `N'-th roots.
coeffs:= CoeffsCyc( z, N );
if coeffs = fail then return fail; fi;
# Get the Zumbroich basis coefficients (basis $B_{n,1}$)
coeffs:= coeffs{ B!.zumbroichbase + 1 };
# Return the list of coefficients.
return coeffs;
end );
#############################################################################
##
#M Quotient( GaussianIntegers, <n>, <m> )
##
InstallMethod( Quotient,
"for Gaussian integers",
IsCollsElmsElms,
[ IsGaussianIntegers, IsCyc, IsCyc ],
function ( GaussianIntegers, x, y )
local q;
if y = 0 then
return fail;
fi;
q := x / y;
if not IsCycInt( q ) then
q := fail;
fi;
return q;
end );
#############################################################################
##
#M StandardAssociateUnit( GaussianIntegers, <x> ) . . for Gaussian integers
##
## The standard associate of <x> is an associated element <y> of <x> that
## lies in the first quadrant of the complex plane.
## That is <y> is that element from `<x> * [1,-1,E(4),-E(4)]' that has
## positive real part and nonnegative imaginary part.
## (This is the generalization of `Abs' (see "Abs") for Gaussian integers.)
##
## This function returns the unit <z> equal to <y> / <x>. The default
## StandardAssociate method then uses this to compute the standard associate.
##
InstallMethod( StandardAssociateUnit,
"for Gaussian integers",
IsCollsElms,
[ IsGaussianIntegers, IsCyc ],
function ( GaussianIntegers, x )
if not IsGaussInt( x ) then
Error( "<x> must lie in <GaussianIntegers>" );
elif IsRat(x) and 0 <= x then
return 1;
elif IsRat(x) then
return -1;
elif 0 < COEFFS_CYC(x)[1] and 0 <= COEFFS_CYC(x)[2] then
return 1;
elif COEFFS_CYC(x)[1] <= 0 and 0 < COEFFS_CYC(x)[2] then
return -E(4);
elif COEFFS_CYC(x)[1] < 0 and COEFFS_CYC(x)[2] <= 0 then
return -1;
else
return E(4);
fi;
end );
#############################################################################
##
#M EuclideanDegree( GaussianIntegers, <n> )
##
InstallMethod( EuclideanDegree,
"for Gaussian integers",
IsCollsElms,
[ IsGaussianIntegers, IsCyc ],
function( GaussianIntegers, x )
if IsGaussInt( x ) then
return x * GaloisCyc( x, -1 );
else
Error( "<x> must lie in <GaussianIntegers>" );
fi;
end );
#############################################################################
##
#M EuclideanRemainder( GaussianIntegers, <n>, <m> )
##
InstallMethod( EuclideanRemainder,
"for Gaussian integers",
IsCollsElmsElms,
[ IsGaussianIntegers, IsCyc, IsCyc ],
function ( GaussianIntegers, x, y )
if IsGaussInt( x ) and IsGaussInt( y ) then
return x - RoundCyc( x/y ) * y;
else
Error( "<x> and <y> must lie in <GaussianIntegers>" );
fi;
end );
#############################################################################
##
#M EuclideanQuotient( GaussianIntegers, <x>, <y> )
##
InstallMethod( EuclideanQuotient,
"for Gaussian integers",
IsCollsElmsElms,
[ IsGaussianIntegers, IsCyc, IsCyc ],
function ( GaussianIntegers, x, y )
if IsGaussInt( x ) and IsGaussInt( y ) then
return RoundCyc( x/y );
else
Error( "<x> and <y> must lie in <GaussianIntegers>" );
fi;
end );
#############################################################################
##
#M QuotientRemainder( GaussianIntegers, <x>, <y> )
##
InstallMethod( QuotientRemainder,
"for Gaussian integers",
IsCollsElmsElms,
[ IsGaussianIntegers, IsCyc, IsCyc ],
function ( GaussianIntegers, x, y )
local q;
if IsGaussInt( x ) and IsGaussInt( y ) then
q := RoundCyc(x/y);
return [ q, x-q*y ];
else
Error( "<x> and <y> must lie in <GaussianIntegers>" );
fi;
end );
#############################################################################
##
#M IsPrime( GaussianIntegers, <n> )
##
InstallMethod( IsPrime,
"for Gaussian integers and integer",
IsCollsElms,
[ IsGaussianIntegers, IsInt ],
function ( GaussianIntegers, x )
return x mod 4 = 3 and IsPrimeInt( x );
end );
InstallMethod( IsPrime,
"for Gaussian integers and cyclotomic",
IsCollsElms,
[ IsGaussianIntegers, IsCyc ],
function ( GaussianIntegers, x )
if IsGaussInt( x ) then
return IsPrimeInt( x * GaloisCyc( x, -1 ) );
else
Error( "<x> must lie in <GaussianIntegers>" );
fi;
end );
#############################################################################
##
#M Factors( GaussianIntegers, <x> )
##
InstallMethod( Factors,
"for Gaussian integers",
IsCollsElms,
[ IsGaussianIntegers, IsCyc ],
function ( GaussianIntegers, x )
local facs, # factors (result)
prm, # prime factors of the norm
tsq; # representation of `prm' as $x^2 + y^2$
# handle trivial cases
if x in [ 0, 1, -1, E(4), -E(4) ] then
return [ x ];
elif not IsGaussInt( x ) then
Error( "<x> must lie in <GaussianIntegers>" );
fi;
# loop over all factors of the norm of x
facs := [];
for prm in PrimeDivisors( EuclideanDegree( GaussianIntegers, x ) ) do
# $p = 2$ and primes $p = 1$ mod 4 split according to $p = x^2 + y^2$
if prm = 2 or prm mod 4 = 1 then
tsq := TwoSquares( prm );
while IsCycInt( x / (tsq[1]+tsq[2]*E(4)) ) do
Add( facs, (tsq[1]+tsq[2]*E(4)) );
x := x / (tsq[1]+tsq[2]*E(4));
od;
while IsCycInt( x / (tsq[2]+tsq[1]*E(4)) ) do
Add( facs, (tsq[2]+tsq[1]*E(4)) );
x := x / (tsq[2]+tsq[1]*E(4));
od;
# primes $p = 3$ mod 4 stay prime
else
while IsCycInt( x / prm ) do
Add( facs, prm );
x := x / prm;
od;
fi;
od;
Assert( 1, x in [ 1, -1, E(4), -E(4) ],
"'Factors' for Gaussian integers: Cofactor must be a unit\n" );
# the first factor takes the unit
facs[1] := x * facs[1];
# return the result
return facs;
end );
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