Quelle padics.gd Sprache: unbekannt

#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Jens Hollmann.
##
## 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 declaration part of the padic numbers.
##

#############################################################################
##
#C IsPadicNumber
##
## <ManSection>
## <Filt Name="IsPadicNumber" Arg='obj' Type='Category'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareCategory( "IsPadicNumber", IsScalar
 and IsAssociativeElement and IsCommutativeElement );

DeclareCategoryCollections( "IsPadicNumber" );
DeclareCategoryCollections( "IsPadicNumberCollection" );

DeclareSynonym( "IsPadicNumberList", IsPadicNumberCollection and IsList );
DeclareSynonym( "IsPadicNumberTable", IsPadicNumberCollColl and IsTable );

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

#############################################################################
##
#C IsPurePadicNumber(<obj>)
##
## <#GAPDoc Label="IsPurePadicNumber">
## <ManSection>
## <Filt Name="IsPurePadicNumber" Arg='obj' Type='Category'/>
##
## <Description>
## The category of pure <M>p</M>-adic numbers.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsPurePadicNumber", IsPadicNumber );

#############################################################################
##
#C IsPurePadicNumberFamily(<fam>)
##
## <#GAPDoc Label="IsPurePadicNumberFamily">
## <ManSection>
## <Filt Name="IsPurePadicNumberFamily" Arg='fam' Type='Category'/>
##
## <Description>
## The family of pure <M>p</M>-adic numbers.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategoryFamily( "IsPurePadicNumber" );

#############################################################################
##
#C IsPadicExtensionNumber(<obj>)
##
## <#GAPDoc Label="IsPadicExtensionNumber">
## <ManSection>
## <Filt Name="IsPadicExtensionNumber" Arg='obj' Type='Category'/>
##
## <Description>
## The category of elements of the extended <M>p</M>-adic field.
## <Example><![CDATA[
## gap> efam:=PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1]);;
## gap> IsPadicExtensionNumber(PadicNumber(efam,7/9));
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsPadicExtensionNumber", IsPadicNumber );

#############################################################################
##
#C IsPadicExtensionNumberFamily(<fam>)
##
## <#GAPDoc Label="IsPadicExtensionNumberFamily">
## <ManSection>
## <Filt Name="IsPadicExtensionNumberFamily" Arg='fam' Type='Category'/>
##
## <Description>
## Family of elements of the extended <M>p</M>-adic field.
## <Example><![CDATA[
## gap> efam:=PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1]);;
## gap> IsPadicExtensionNumberFamily(efam);
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategoryFamily( "IsPadicExtensionNumber" );

#############################################################################
##
#O Valuation( <obj> )
##
## <#GAPDoc Label="Valuation">
## <ManSection>
## <Oper Name="Valuation" Arg='obj'/>
##
## <Description>
## The valuation is the <M>p</M>-part of the <M>p</M>-adic number.
## See also <Ref Func="PValuation"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "Valuation", [ IsObject ] );

#############################################################################
##
#O PadicNumber( <fam>, <rat> )
#O PadicNumber( <purefam>,<list>)
#O PadicNumber( <extfam>,<list>)
##
## <#GAPDoc Label="PadicNumber">
## <ManSection>
## <Oper Name="PadicNumber" Arg='fam, rat'
## Label="for a p-adic extension family and a rational"/>
## <Oper Name="PadicNumber" Arg='purefam, list'
## Label="for a pure p-adic numbers family and a list"/>
## <Oper Name="PadicNumber" Arg='extfam, list'
## Label="for a p-adic extension family and a list"/>
##
## <Description>
## (see also <Ref Oper="PadicNumber" Label="for pure padics"/>).
## 
## <Ref Oper="PadicNumber"
## Label="for a p-adic extension family and a rational"/>
## creates a <M>p</M>-adic number in the
## <M>p</M>-adic numbers family <A>fam</A>.
## The first form returns the <M>p</M>-adic number corresponding to the
## rational <A>rat</A>.
## 
## The second form takes a pure <M>p</M>-adic numbers family <A>purefam</A>
## and a list <A>list</A> of length two, and returns the number
## <M>p</M><C>^</C><A>list</A><C>[1] * </C><A>list</A><C>[2]</C>.
## It must be guaranteed that no entry of <A>list</A><C>[2]</C> is
## divisible by the prime <M>p</M>.
## (Otherwise precision will get lost.)
## 
## The third form creates a number in the family <A>extfam</A> of a
## <M>p</M>-adic extension.
## The second argument must be a list <A>list</A> of length two such that
## <A>list</A><C>[2]</C> is the list of coefficients w.r.t. the basis
## <M>\{ 1, \ldots, x^{{f-1}} \cdot y^{{e-1}} \}</M> of the extended
## <M>p</M>-adic field and <A>list</A><C>[1]</C> is a common <M>p</M>-part
## of all these coefficients.
## 
## <M>p</M>-adic numbers admit the usual field operations.
## <Example><![CDATA[
## gap> efam:=PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1]);;
## gap> PadicNumber(efam,7/9);
## padic(120(3),0(3))
## ]]></Example>
## 
## <E>A word of warning:</E>
## 
## Depending on the actual representation of quotients, precision may seem
## to <Q>vanish</Q>.
## For example in <C>PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1])</C>
## the number <C>(1.2000, 0.1210)(3)</C> can be represented as
## <C>[ 0, [ 1.2000, 0.1210 ] ]</C> or as <C>[ -1, [ 12.000, 1.2100 ] ]</C>
## (here the coefficients have to be multiplied by <M>p^{{-1}}</M>).
## 
## So there may be a number <C>(1.2, 2.2)(3)</C> which seems to have
## only two digits of precision instead of the declared 5.
## But internally the number is stored as <C>[ -3, [ 0.0012, 0.0022 ] ]</C>
## and so has in fact maximum precision.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "PadicNumber", [ IsPadicNumberFamily, IsObject ] );

#############################################################################
##
#O ShiftedPadicNumber( <padic>, <int> )
##
## <#GAPDoc Label="ShiftedPadicNumber">
## <ManSection>
## <Oper Name="ShiftedPadicNumber" Arg='padic, int'/>
##
## <Description>
## <Ref Oper="ShiftedPadicNumber"/> takes a <M>p</M>-adic number
## <A>padic</A> and an integer <A>shift</A>
## and returns the <M>p</M>-adic number <M>c</M>,
## that is <A>padic</A> <C>*</C> <M>p</M><C>^</C><A>shift</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "ShiftedPadicNumber", [ IsPadicNumber, IsInt ] );

#############################################################################
##
#O PurePadicNumberFamily( , <precision> )
##
## <#GAPDoc Label="PurePadicNumberFamily">
## <ManSection>
## <Func Name="PurePadicNumberFamily" Arg='p, precision'/>
##
## <Description>
## returns the family of pure <M>p</M>-adic numbers over the
## prime <A>p</A> with <A>precision</A> <Q>digits</Q>. That is to say, the approximate value
## will differ from the correct value by a multiple of <M>p^{digits}</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PurePadicNumberFamily" );

#############################################################################
##
#F PadicExtensionNumberFamily( , <precision>, <unram>, <ram> )
##
## <#GAPDoc Label="PadicExtensionNumberFamily">
## <ManSection>
## <Func Name="PadicExtensionNumberFamily" Arg='p, precision, unram, ram'/>
##
## <Description>
## An extended <M>p</M>-adic field <M>L</M> is given by two polynomials
## <M>h</M> and <M>g</M> with coefficient lists <A>unram</A> (for the
## unramified part) and <A>ram</A> (for the ramified part).
## Then <M>L</M> is isomorphic to <M>Q_p[x,y]/(h(x),g(y))</M>.
## 
## This function takes the prime number <A>p</A> and the two coefficient
## lists <A>unram</A> and <A>ram</A> for the two polynomials.
## The polynomial given by the coefficients in <A>unram</A> must be a
## cyclotomic polynomial and the polynomial given by <A>ram</A> must be
## either an Eisenstein polynomial or <M>1+x</M>.
## <E>This is not checked by &GAP;.</E>
## 
## Every number in <M>L</M> is represented as a coefficient list w. r. t.
## the basis <M>\{ 1, x, x^2, \ldots, y, xy, x^2 y, \ldots \}</M>
## of <M>L</M>.
## The integer <A>precision</A> is the number of <Q>digits</Q> that all the
## coefficients have.
## 
## <E>A general comment:</E>
## 
## The polynomials with which <Ref Func="PadicExtensionNumberFamily"/> is
## called define an extension of <M>Q_p</M>.
## It must be ensured that both polynomials are really irreducible over
## <M>Q_p</M>!
## For example <M>x^2+x+1</M> is <E>not</E> irreducible over <M>Q_p</M>.
## Therefore the <Q>extension</Q>
## <C>PadicExtensionNumberFamily(3, 4, [1,1,1], [1,1])</C> contains
## non-invertible <Q>pseudo-p-adic numbers</Q>.
## Conversely, if an <Q>extension</Q> contains noninvertible elements
## then one of the defining polynomials was not irreducible.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PadicExtensionNumberFamily" );

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