#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This files's authors include Thomas Breuer.
##
## 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 declarations 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.
##
#############################################################################
##
#F IsGaussInt( <x> ) . . . . . . . . test if an object is a Gaussian integer
##
## <#GAPDoc Label="IsGaussInt">
## <ManSection>
## <Func Name="IsGaussInt" Arg='x'/>
##
## <Description>
## <Ref Func="IsGaussInt"/> returns <K>true</K> if the object <A>x</A> is
## a Gaussian integer (see <Ref Var="GaussianIntegers"/>),
## and <K>false</K> otherwise.
## Gaussian integers are of the form <M>a + b</M><C>*E(4)</C>,
## where <M>a</M> and <M>b</M> are integers.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "IsGaussInt" );
#############################################################################
##
#F IsGaussRat( <x> ) . . . . . . . test if an object is a Gaussian rational
##
## <#GAPDoc Label="IsGaussRat">
## <ManSection>
## <Func Name="IsGaussRat" Arg='x'/>
##
## <Description>
## <Ref Func="IsGaussRat"/> returns <K>true</K> if the object <A>x</A> is
## a Gaussian rational (see <Ref Var="GaussianRationals"/>),
## and <K>false</K> otherwise.
## Gaussian rationals are of the form <M>a + b</M><C>*E(4)</C>,
## where <M>a</M> and <M>b</M> are rationals.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "IsGaussRat" );
#############################################################################
##
#C IsGaussianIntegers( <obj> )
##
## <#GAPDoc Label="IsGaussianIntegers">
## <ManSection>
## <Filt Name="IsGaussianIntegers" Arg='obj' Type='Category'/>
##
## <Description>
## is the defining category for the domain <Ref Var="GaussianIntegers"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsGaussianIntegers", IsEuclideanRing and IsFLMLOR
and IsFiniteDimensional );
#############################################################################
##
#V GaussianIntegers . . . . . . . . . . . . . . . ring of Gaussian integers
##
## <#GAPDoc Label="GaussianIntegers">
## <ManSection>
## <Var Name="GaussianIntegers"/>
##
## <Description>
## <Ref Var="GaussianIntegers"/> is the ring <M>&ZZ;[\sqrt{{-1}}]</M>
## of Gaussian integers.
## This is a subring of the cyclotomic field
## <Ref Var="GaussianRationals"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalName( "GaussianIntegers");
