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  68 p-adic Numbers (preliminary)
  
  In this chapter p is always a (fixed) prime integer.
  
  The  p-adic  numbers  Q_p  are  the  completion of the rational numbers with
  respect to the valuation ν_p( p^v ⋅ a / b) = v if p divides neither a nor b.
  They  form  a  field  of  characteristic  0  which  nevertheless  shows some
  behaviour of the finite field with p elements.
  
  A  p-adic  numbers can be represented by a p-adic expansion which is similar
  to  the  decimal  expansion  used  for  the  reals (but written from left to
  right).  So  for  example if p = 2, the numbers 1, 2, 3, 4, 1/2, and 4/5 are
  represented  as  1(2),  0.1(2),  1.1(2),  0.01(2),  10(2),  and the infinite
  periodic   expansion   0.010110011001100...(2).   p-adic   numbers   can  be
  approximated  by  ignoring  higher  powers  of p, so for example with only 2
  digits accuracy 4/5 would be approximated as 0.01(2). This is different from
  the  decimal approximation of real numbers in that p-adic approximation is a
  ring  homomorphism  on  the  subrings  of  p-adic numbers whose valuation is
  bounded  from  below  so  that rounding errors do not increase with repeated
  calculations.
  
  In  GAP,  p-adic  numbers  are  always represented by such approximations. A
  family  of  approximated  p-adic  numbers  consists of p-adic numbers with a
  fixed  prime p and a certain precision, and arithmetic with these numbers is
  done with this precision.
  
  
  68.1 Pure p-adic Numbers
  
  Pure p-adic numbers are the p-adic numbers described so far.
  
  68.1-1 PurePadicNumberFamily
  
  PurePadicNumberFamily( p, precision )  function
  
  returns  the  family  of pure p-adic numbers over the prime p with precision
  digits.  That  is to say, the approximate value will differ from the correct
  value by a multiple of p^digits.
  
  68.1-2 PadicNumber
  
  PadicNumber( fam, rat )  operation
  
  returns  the  element  of the p-adic number family fam that approximates the
  rational number rat.
  
  p-adic numbers allow the usual operations for fields.
  
    Example  
    gap> fam:=PurePadicNumberFamily(2,20);;
    gap> a:=PadicNumber(fam,4/5);
    0.010110011001100110011(2)
    gap> fam:=PurePadicNumberFamily(2,3);;
    gap> a:=PadicNumber(fam,4/5);
    0.0101(2)
    gap> 3*a;
    0.0111(2)
    gap> a/2;
    0.101(2)
    gap> a*10;
    0.001(2)
  
  
  See PadicNumber (68.2-2) for other methods for PadicNumber.
  
  68.1-3 Valuation
  
  Valuation( obj )  operation
  
  The  valuation  is  the  p-part  of  the  p-adic number. See also PValuation
  (15.7-1).
  
  68.1-4 ShiftedPadicNumber
  
  ShiftedPadicNumber( padic, int )  operation
  
  ShiftedPadicNumber  takes  a  p-adic  number  padic and an integer shift and
  returns the p-adic number c, that is padic * p^shift.
  
  68.1-5 IsPurePadicNumber
  
  IsPurePadicNumber( obj )  Category
  
  The category of pure p-adic numbers.
  
  68.1-6 IsPurePadicNumberFamily
  
  IsPurePadicNumberFamily( fam )  Category
  
  The family of pure p-adic numbers.
  
  
  68.2 Extensions of the p-adic Numbers
  
  The  usual Kronecker construction with an irreducible polynomial can be used
  to  construct  extensions of the p-adic numbers. Let L be such an extension.
  Then there is a subfield K < L such that K is an unramified extension of the
  p-adic numbers and L/K is purely ramified.
  
  (For  an  explanation of ramification see for example [Neu92, Section II.7],
  or  another  book on algebraic number theory. Essentially, an extension L of
  the  p-adic  numbers generated by a rational polynomial f is unramified if f
  remains  squarefree  modulo  p  and  is  completely ramified if modulo p the
  polynomial  f is a power of a linear factor while remaining irreducible over
  the p-adic numbers.)
  
  The  representation of extensions of p-adic numbers in GAP uses the subfield
  K.
  
  68.2-1 PadicExtensionNumberFamily
  
  PadicExtensionNumberFamily( p, precision, unram, ram )  function
  
  An  extended  p-adic  field  L  is  given  by  two  polynomials h and g with
  coefficient  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 must
  be either an Eisenstein polynomial or 1+x. This is not checked by GAP.
  
  Every  number in L is represented as a coefficient list w. r. t. the basis {
  1,  x,  x^2,  ...,  y,  xy,  x^2 y, ... } of L. The integer precision is the
  number of digits that all the coefficients have.
  
  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 then one of the defining polynomials was not
  irreducible.
  
  68.2-2 PadicNumber
  
  PadicNumber( fam, rat )  operation
  PadicNumber( purefam, list )  operation
  PadicNumber( extfam, list )  operation
  
  (see also PadicNumber (68.1-2)).
  
  PadicNumber  creates  a  p-adic number in the p-adic numbers family fam. The
  first form returns the p-adic number corresponding to the rational rat.
  
  The  second  form takes a pure p-adic numbers family purefam and a list list
  of  length  two,  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 form creates a number in the family extfam of a p-adic extension.
  The  second  argument must be a list list of length two such that list[2] is
  the  list  of coefficients w.r.t. the basis { 1, ..., x^{f-1} ⋅ y^{e-1} } of
  the  extended  p-adic  field  and  list[1]  is  a common p-part of all these
  coefficients.
  
  p-adic numbers admit the usual field operations.
  
    Example  
    gap> efam:=PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1]);;
    gap> PadicNumber(efam,7/9);
    padic(120(3),0(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.
  
  68.2-3 IsPadicExtensionNumber
  
  IsPadicExtensionNumber( obj )  Category
  
  The category of elements of the extended p-adic field.
  
    Example  
    gap>  efam:=PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1]);;
    gap> IsPadicExtensionNumber(PadicNumber(efam,7/9));
    true
  
  
  68.2-4 IsPadicExtensionNumberFamily
  
  IsPadicExtensionNumberFamily( fam )  Category
  
  Family of elements of the extended p-adic field.
  
    Example  
    gap> efam:=PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1]);;
    gap> IsPadicExtensionNumberFamily(efam);
    true