Quelle zmodnz.gd Sprache: unbekannt

#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file'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 the design of the rings $Z / n Z$ and their elements.
##
## The ordering of elements for nonprime $n$ is defined by the ordering of
## the representatives.
## For primes smaller than `MAXSIZE_GF_INTERNAL', the ordering of the
## internal finite field elements must be respected, for larger primes
## again the ordering of representatives is chosen.
##

#############################################################################
##
#C IsZmodnZObj( <obj> )
#C IsZmodnZObjNonprime( <obj> )
#C IsZmodpZObj( <obj> )
#C IsZmodpZObjSmall( <obj> )
#C IsZmodpZObjLarge( <obj> )
##
## <#GAPDoc Label="IsZmodnZObj">
## <ManSection>
## <Filt Name="IsZmodnZObj" Arg='obj' Type='Category'/>
## <Filt Name="IsZmodnZObjNonprime" Arg='obj' Type='Category'/>
## <Filt Name="IsZmodpZObj" Arg='obj' Type='Category'/>
## <Filt Name="IsZmodpZObjSmall" Arg='obj' Type='Category'/>
## <Filt Name="IsZmodpZObjLarge" Arg='obj' Type='Category'/>
##
## <Description>
## The elements in the rings <M>Z / n Z</M> are in the category
## <Ref Filt="IsZmodnZObj"/>.
## If <M>n</M> is a prime then the elements are of course also in the
## category <Ref Filt="IsFFE"/>,
## otherwise they are in <Ref Filt="IsZmodnZObjNonprime"/>.
## <Ref Filt="IsZmodpZObj"/> is an abbreviation of
## <C>IsZmodnZObj and IsFFE</C>.
## This category is the disjoint union of <Ref Filt="IsZmodpZObjSmall"/> and
## <Ref Filt="IsZmodpZObjLarge"/>, the former containing all elements with
## <M>n</M> at most <C>MAXSIZE_GF_INTERNAL</C>.
## 
## The reasons to distinguish the prime case from the nonprime case are
## <List>
## <Item>
## that objects in <Ref Filt="IsZmodnZObjNonprime"/> have an external
## representation (namely the residue in the range
## <M>[ 0, 1, \ldots, n-1 ]</M>),
## </Item>
## <Item>
## that the comparison of elements can be defined as comparison of the
## residues, and
## </Item>
## <Item>
## that the elements lie in a family of type
## <C>IsZmodnZObjNonprimeFamily</C>
## (note that for prime <M>n</M>, the family must be an
## <C>IsFFEFamily</C>).
## </Item>
## </List>
## 
## The reasons to distinguish the small and the large case are
## that for small <M>n</M> the elements must be compatible with the internal
## representation of finite field elements, whereas we are free to define
## comparison as comparison of residues for large <M>n</M>.
## 
## Note that we <E>cannot</E> claim that every finite field element of
## degree 1 is in <Ref Filt="IsZmodnZObj"/>, since finite field elements in
## internal representation may not know that they lie in the prime field.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsZmodnZObj", IsScalar and IsAssociativeElement
 and IsCommutativeElement and IsAdditivelyCommutativeElement );
DeclareCategory( "IsZmodnZObjNonprime", IsZmodnZObj );
DeclareSynonym( "IsZmodpZObj", IsZmodnZObj and IsFFE );
DeclareSynonym( "IsZmodpZObjSmall", IsZmodpZObj and IsLogOrderedFFE );
DeclareSynonym( "IsZmodpZObjLarge", IsZmodpZObj and IsLexOrderedFFE );

#############################################################################
##
#C IsZmodnZObjNonprimeFamily( <obj> )
##
## <ManSection>
## <Filt Name="IsZmodnZObjNonprimeFamily" Arg='obj' Type='Category'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareCategoryFamily( "IsZmodnZObjNonprime" );

#############################################################################
##
#C IsZmodnZObjNonprimeCollection( <obj> )
#C IsZmodnZObjNonprimeCollColl( <obj> )
#C IsZmodnZObjNonprimeCollCollColl( <obj> )
##
## <ManSection>
## <Filt Name="IsZmodnZObjNonprimeCollection" Arg='obj' Type='Category'/>
## <Filt Name="IsZmodnZObjNonprimeCollColl" Arg='obj' Type='Category'/>
## <Filt Name="IsZmodnZObjNonprimeCollCollColl" Arg='obj' Type='Category'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareCategoryCollections( "IsZmodnZObjNonprime" );
DeclareCategoryCollections( "IsZmodnZObjNonprimeCollection" );
DeclareCategoryCollections( "IsZmodnZObjNonprimeCollColl" );

#############################################################################
##
#M IsFinite( <R> ) . . . . . . . . . . . . . . . . method for full ring Z/nZ
##
InstallTrueMethod( IsFinite,
 IsZmodnZObjNonprimeCollection and IsDuplicateFree );

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##
#M IsEuclideanRing( <R> ) . . . . . . . . . . . . method for full ring Z/nZ
##
InstallTrueMethod(IsEuclideanRing, IsZmodnZObjNonprimeCollection and
 IsWholeFamily and IsRing);

#############################################################################
##
#V Z_MOD_NZ
##
## <ManSection>
## <Var Name="Z_MOD_NZ"/>
##
## <Description>
## is a list of length 2, the first containing at position <A>i</A> the
## <A>i</A>-th value <A>n</A> for that <C>ZmodnZ( <A>n</A> )</C> is stored,
## and the second containing this ring at position <A>i</A>.
## </Description>
## </ManSection>
##
BindGlobal( "Z_MOD_NZ", NEW_SORTED_CACHE(true) );

#############################################################################
##
#F ZmodnZ( <n> )
#F ZmodpZ( )
#F ZmodpZNC( )
##
## <#GAPDoc Label="ZmodnZ">
## <ManSection>
## <Func Name="ZmodnZ" Arg='n'/>
## <Func Name="ZmodpZ" Arg='p'/>
## <Func Name="ZmodpZNC" Arg='p'/>
##
## <Description>
## <Ref Func="ZmodnZ"/> returns a ring <M>R</M> isomorphic to the residue
## class ring of the integers modulo the ideal generated by <A>n</A>.
## The element corresponding to the residue class of the integer <M>i</M>
## in this ring can be obtained by <C>i * One( R )</C>,
## and a representative of the residue class corresponding to the element
## <M>x \in R</M> can be computed by <C>Int</C><M>( x )</M>.
## 
## <Index Subkey="Integers">mod</Index>
## <C>ZmodnZ( <A>n</A> )</C> is equal to <C>Integers mod <A>n</A></C>.
## 
## <Ref Func="ZmodpZ"/> does the same if the argument <A>p</A> is a prime
## integer, additionally the result is a field.
## <Ref Func="ZmodpZNC"/> omits the check whether <A>p</A> is a prime.
## 
## Each ring returned by these functions contains the whole family of its
## elements
## if <A>n</A> is not a prime, and is embedded into the family of finite
## field elements of characteristic <A>n</A> if <A>n</A> is a prime.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ZmodnZ" );
DeclareGlobalFunction( "ZmodpZ" );
DeclareGlobalFunction( "ZmodpZNC" );

#############################################################################
##
#O ZmodnZObj( <Fam>, <r> )
#O ZmodnZObj( <r>, <n> )
##
## <#GAPDoc Label="ZmodnZObj">
## <ManSection>
## <Oper Name="ZmodnZObj"
## Arg='Fam, r' Label="for a residue class family and integer"/>
## <Oper Name="ZmodnZObj" Arg='r, n' Label="for two integers"/>
##
## <Description>
## If the first argument is a residue class family <A>Fam</A> then
## <Ref Oper="ZmodnZObj" Label="for a residue class family and integer"/>
## returns the element in <A>Fam</A> whose coset is represented by the
## integer <A>r</A>.
## 
## If the two arguments are an integer <A>r</A> and a positive integer
## <A>n</A> then <Ref Oper="ZmodnZObj" Label="for two integers"/>
## returns the element in <C>ZmodnZ( <A>n</A> )</C>
## (see <Ref Func="ZmodnZ"/>) whose coset is represented by the integer
## <A>r</A>.
## 
## <Example><![CDATA[
## gap> r:= ZmodnZ(15);
## (Integers mod 15)
## gap> fam:=ElementsFamily(FamilyObj(r));;
## gap> a:= ZmodnZObj(fam,9);
## ZmodnZObj( 9, 15 )
## gap> a+a;
## ZmodnZObj( 3, 15 )
## gap> Int(a+a);
## 3
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "ZmodnZObj", [ IsZmodnZObjNonprimeFamily, IsInt ] );
DeclareOperation( "ZmodnZObj", [ IsInt, IsPosInt ] );
DeclareSynonym( "ZmodpZObj", ZmodnZObj );

#############################################################################
##
#A ModulusOfZmodnZObj( <obj> )
##
## <ManSection>
## <Attr Name="ModulusOfZmodnZObj" Arg='obj'/>
##
## <Description>
## For an element <A>obj</A> in a residue class ring of integers modulo
## <M>n</M> (see <Ref Filt="IsZmodnZObj"/>),
## <Ref Attr="ModulusOfZmodnZObj"/> returns the positive integer <M>n</M>.
##
## Deprecated, use <Ref Attr="Characteristic"/> instead.
## </Description>
## </ManSection>
##
DeclareAttribute( "ModulusOfZmodnZObj", IsZmodnZObj );

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##
#F EnumeratorOfZmodnZ( <R> ). . . . . . . . . . . . . enumerator for Z / n Z
##
## <ManSection>
## <Func Name="EnumeratorOfZmodnZ" Arg='R'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "EnumeratorOfZmodnZ" );

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##
#M IsFinite( <zmodnz-mat-grp> )
##
## *NOTE*: The following implication only holds if there are no infinite
## dimensional matrices.
##
InstallTrueMethod( IsFinite,
 IsZmodnZObjNonprimeCollCollColl and IsRingElementCollCollColl
 and IsGroup
 and IsFinitelyGeneratedGroup );

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