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%A padics.msk GAP documentation Alexander Hulpke
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%A @(#)$Id: padics.msk,v 1.6 2002/04/15 10:02:31 sal Exp $
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%Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland
%Y Copyright (C) 2002 The GAP Group
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\PreliminaryChapter{p-adic Numbers}
In this chapter $p$ is always a (fixed) prime.
The $p$-adic numbers $Q_p$ are the completion of the rational numbers with
respect to the valuation $\nu_p(p^v\frac{a}{b})=v$ 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 approximated 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 $1$,$2$,$3$,$4$,$\frac{1}{2}$ and $\frac{4}{5}$ are
represented as $1(2)$, $0.1(2)$, $1.1(2)$, $0.01(2)$, $10(2)$ and
$0.0101(2)$. Approximation means to ignore powers of $p$, so for example
with only 2 digits accuracy $\frac{4}{5}$ would be approximated as
$0.01(2)$.
The important difference to the decimal approximation is that
$p$-adic approximation is a ring homomorphism on the subrings of $p$-adic
numbers whose valuation is bounded from below.
In {\GAP}, $p$-adic numbers are represented by approximations. A family of
(approximated) $p$-adic numbers consists of $p$-adic numbers with a certain
precision and arithmetic with these numbers is done with this precision.
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\Section{Pure p-adic Numbers}
Pure $p$-adic numbers are the $p$-adic numbers described so far.
\>PurePadicNumberFamily( <p>, <precision> ) O
returns the family of pure $p$-adic numbers over the
prime <p> with <precision> ``digits''.
\>PadicNumber(<fam>,<rat>)!{for pure padics}
returns the element of the $p$-adic number family <fam> that is used to
represent the rational number <rat>.
$p$-adic numbers allow the usual operations for fields.
\beginexample
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)
\endexample
\>Valuation( <obj> ) O
The Valuation is the $p$-part of the $p$-adic number.
\>ShiftedPadicNumber( <padic>, <int> ) O
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.
\>IsPurePadicNumber( <obj> ) C
\>IsPurePadicNumberFamily( <fam> ) C
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\Section{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 \cite{neukirch},
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 this
subfield.
\>PadicExtensionNumberFamily( <p>, <precision>, <unram>, <ram> ) F
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.
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.
\>PadicNumber( <fam>, <rat> ) O
\>PadicNumber( <purefam>, <list> ) O
\>PadicNumber( <extfam>, <list> ) O
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.
\beginexample
gap> efam:=PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1]);;
gap> PadicNumber(efam,7/9);
padic(120(3),0(3))
\endexample
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.
\>IsPadicExtensionNumber( <obj> ) C
\>IsPadicExtensionNumberFamily( <fam> ) C
