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#############################################################################
##
#W padics.gd GAP Library Jens Hollmann
##
#H @(#)$Id: padics.gd,v 4.18 2002/04/15 10:05:11 sal Exp $
##
#Y Copyright (C) 1997, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains the declaration part of the padic numbers.
##
Revision.padics_gd :=
"@(#)$Id: padics.gd,v 4.18 2002/04/15 10:05:11 sal Exp $";
#############################################################################
##
#C IsPadicNumber
##
DeclareCategory( "IsPadicNumber", IsScalar
and IsAssociativeElement and IsCommutativeElement );
DeclareCategoryCollections( "IsPadicNumber" );
DeclareCategoryCollections( "IsPadicNumberCollection" );
DeclareSynonym( "IsPadicNumberList", IsPadicNumberCollection and IsList );
DeclareSynonym( "IsPadicNumberTable", IsPadicNumberCollColl and IsTable );
#############################################################################
##
#C IsPadicNumberFamily
##
DeclareCategoryFamily( "IsPadicNumber" );
#############################################################################
##
#C IsPurePadicNumber(<obj>)
##
DeclareCategory( "IsPurePadicNumber", IsPadicNumber );
#############################################################################
##
#C IsPurePadicNumberFamily(<fam>)
##
DeclareCategoryFamily( "IsPurePadicNumber" );
#############################################################################
##
#C IsPadicExtensionNumber(<obj>)
##
DeclareCategory( "IsPadicExtensionNumber", IsPadicNumber );
#############################################################################
##
#C IsPadicExtensionNumberFamily(<fam>)
##
DeclareCategoryFamily( "IsPadicExtensionNumber" );
#############################################################################
##
#O Valuation( <obj> )
##
## The Valuation is the $p$-part of the $p$-adic number.
DeclareOperation( "Valuation", [ IsObject ] );
#############################################################################
##
#O PadicNumber( <fam>, <rat> )
#O PadicNumber( <purefam>,<list>)
#O PadicNumber( <extfam>,<list>)
##
## create a $p$-adic number in the $p$-adic numbers family <fam>. The
## first usage returns the $p$-adic number corresponding to the rational
## <rat>.
##
## The second usage takes a pure $p$-adic numbers family <purefam> and a
## list <list> of length 2 and returns the number `p^<list>[1] *
## <list>[2]'. It must be guaranteed that no entry of list[2] is
## divisible by the prime p. (Otherwise precision will get lost.)
##
## The third usage creates a number in the family <extfam> of a $p$-adic
## extension. The second entry must be a list <L> of length 2 such that
## <list>[2] is the list of coeff. for the basis
## $\{1,\ldots,x^{f-1}\cdot y^{e-1}\}$ of the extended $p$-adic field and
## <list>[1] is a common $p$-part of all the coeff.
##
## $p$-adic numbers allow the usual field operations.
##
DeclareOperation( "PadicNumber", [ IsPadicNumberFamily, IsObject ] );
#############################################################################
##
#O ShiftedPadicNumber( <padic>, <int> )
##
## ShiftedPadicNumber takes a $p$-adic number <padic> and an integer <shift>
## and returns the $p$-adic number $c$, that is `<padic>* p^<shift>'. The
## <shift> is just added to the $p$-part.
##
DeclareOperation( "ShiftedPadicNumber", [ IsPadicNumber, IsInt ] );
#############################################################################
##
#O PurePadicNumberFamily( <p>, <precision> )
##
## returns the family of pure $p$-adic numbers over the
## prime <p> with <precision> ``digits''.
##
DeclareGlobalFunction( "PurePadicNumberFamily" );
#############################################################################
##
#F PadicExtensionNumberFamily( <p>, <precision>, <unram>, <ram> )
##
## An extended $p$-adic field $L$ is given by two polynomials h and g with
## coeff.-lists <unram> (for the unramified part) and <ram> (for the
## ramified part). Then $L$ is isomorphic to $Q_p[x,y]/(h(x),g(y))$.
##
## This function takes the prime number <p> and the two coefficient lists
## <unram> and <ram> for the two polynomials. The polynomial given by the
## coefficients in <unram> must be a cyclotomic polynomial and the
## polynomial given by <ram> an Eisenstein-polynomial (or 1+x). *This is
## not checked by {\sf GAP}.*
##
## Every number out of $L$ is represented as a coeff.-list for
## the basis $\{1,x,x^2,\ldots,y,xy,x^2y,\ldots\}$ of $L$. The integer
## <precision> is the number of ``digits'' that all the coefficients have.
##
DeclareGlobalFunction( "PadicExtensionNumberFamily" );
#2
## A general comment: the polynomials with which
## `PadicExtensionNumberFamily' is called define an extension of $Q_p$. It
## must be ensured that both polynomials are really irreducible over
## $Q_p$! For example x^2+x+1 is *not* irreducible over Q_p. Therefore the
## ``extension'' PadicExtensionNumberFamily(3, 4, [1,1,1], [1,1]) contains
## non-invertible ``pseudo-$p$-adic numbers''. Conversely, if an
## ``extension'' contains noninvertible elements one of the polynomials was
## not irreducible.
##
#3
## A word of warning:
## Depending on the actual representation of quotients, precision may seem
## to ``vanish''.
## For example in PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1]) the
## number (1.2000, 0.1210)(3) can be represented as `[ 0, [ 1.2000, 0.1210
## ] ]' or as `[-1, [ 12.000, 1.2100 ] ]' (here the coefficients have to be
## multiplied by $p^{-1}$).
##
## So there may be a number (1.2, 2.2)(3) which seems to have only two
## digits of precision instead of the declared 5. But internally the number
## is stored as `[-3, [ 0.0012, 0.0022 ] ]' and so has in fact maximum
## precision.
#############################################################################
##
#E padics.gd . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
##
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