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<h1>66 p-adic Numbers</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP066.htm#SECT001">Pure p-adic Numbers</a>
<li> <A HREF="CHAP066.htm#SECT002">Extensions of the p-adic Numbers</a>
</ol><p>
<p>
In this chapter <i>p</i> is always a (fixed) prime.
<p>
The <i>p</i>-adic numbers <i>Q</i><sub><i>p</i></sub> are the completion of the rational numbers with
respect to the valuation ν<sub><i>p</i></sub>(<i>p</i><sup><i>v</i></sup>[(<i>a</i>)/(<i>b</i>)])=<i>v</i> if <i>p</i> divides neither
<i>a</i> nor <i>b</i>. They form a field of characteristic 0 which nevertheless shows
some behaviour of the finite field with <i>p</i> elements.
<p>
A <i>p</i>-adic numbers can be approximated by a ``<i>p</i>-adic expansion'' which is
similar to the decimal expansion used for the reals (but written from left
to right). So for example if
<i>p</i>=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
0·0101(2). Approximation means to ignore powers of <i>p</i>, so for example
with only 2 digits accuracy [4/5] would be approximated as
0·01(2).
The important difference to the decimal approximation is that
<i>p</i>-adic approximation is a ring homomorphism on the subrings of <i>p</i>-adic
numbers whose valuation is bounded from below.
<p>
In <font face="Gill Sans,Helvetica,Arial">GAP</font>, <i>p</i>-adic numbers are represented by approximations. A family of
(approximated) <i>p</i>-adic numbers consists of <i>p</i>-adic numbers with a certain
precision and arithmetic with these numbers is done with this precision.
<p>
<p>
<h2><a name="SECT001">66.1 Pure p-adic Numbers</a></h2>
<p><p>
Pure <i>p</i>-adic numbers are the <i>p</i>-adic numbers described so far.
<p>
<a name = "SSEC001.1"></a>
<li><code>PurePadicNumberFamily( </code><var>p</var><code>, </code><var>precision</var><code> ) O</code>
<p>
returns the family of pure <i>p</i>-adic numbers over the
prime <var>p</var> with <var>precision</var> ``digits''.
<p>
<a name = "SSEC001.2"></a>
<li><code>PadicNumber(</code><var>fam</var><code>,</code><var>rat</var><code>)</code>
<p>
returns the element of the <i>p</i>-adic number family <var>fam</var> that is used to
represent the rational number <var>rat</var>.
<p>
<i>p</i>-adic numbers allow the usual operations for fields.
<p>
<pre>
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)
</pre>
<p>
<a name = "SSEC001.3"></a>
<li><code>Valuation( </code><var>obj</var><code> ) O</code>
<p>
The Valuation is the <i>p</i>-part of the <i>p</i>-adic number.
<p>
<a name = "SSEC001.4"></a>
<li><code>ShiftedPadicNumber( </code><var>padic</var><code>, </code><var>int</var><code> ) O</code>
<p>
ShiftedPadicNumber takes a <i>p</i>-adic number <var>padic</var> and an integer <var>shift</var>
and returns the <i>p</i>-adic number <i>c</i>, that is <code></code><var>padic</var><code>* p^</code><var>shift</var><code></code>. The
<var>shift</var> is just added to the <i>p</i>-part.
<p>
<a name = "SSEC001.5"></a>
<li><code>IsPurePadicNumber( </code><var>obj</var><code> ) C</code>
<p>
<a name = "SSEC001.6"></a>
<li><code>IsPurePadicNumberFamily( </code><var>fam</var><code> ) C</code>
<p>
<p>
<h2><a name="SECT002">66.2 Extensions of the p-adic Numbers</a></h2>
<p><p>
The usual Kronecker construction with an irreducible polynomial can be used
to construct extensions of the <i>p</i>-adic numbers.
Let <i>L</i> be such an
extension. Then there is a subfield <i>K</i> < <i>L</i> such that <i>K</i> is an unramified
extension of the <i>p</i>-adic numbers and <i>L</i>/<i>K</i> is purely ramified.
(For an explanation of ``ramification'' see for example <a href="biblio.htm#neukirch"><cite>neukirch</cite></a>,
section II.7 or another book on algebraic number theory. Essentially, an
extension <i>L</i> of the <i>p</i>-adic numbers generated
by a rational polynomial <i>f</i> is unramified if <i>f</i> remains squarefree modulo
<i>p</i> and is completely ramified if modulo <i>p</i> the polynomial <i>f</i> is a power of
a linear factor while remaining irreducible over the <i>p</i>-adic numbers.)
The representation of extensions of <i>p</i>-adic numbers in <font face="Gill Sans,Helvetica,Arial">GAP</font> uses this
subfield.
<p>
<a name = "SSEC002.1"></a>
<li><code>PadicExtensionNumberFamily( </code><var>p</var><code>, </code><var>precision</var><code>, </code><var>unram</var><code>, </code><var>ram</var><code> ) F</code>
<p>
An extended <i>p</i>-adic field <i>L</i> is given by two polynomials h and g with
coeff.-lists <var>unram</var> (for the unramified part) and <var>ram</var> (for the
ramified part). Then <i>L</i> is isomorphic to <i>Q</i><sub><i>p</i></sub>[<i>x</i>,<i>y</i>]/(<i>h</i>(<i>x</i>),<i>g</i>(<i>y</i>)).
<p>
This function takes the prime number <var>p</var> and the two coefficient lists
<var>unram</var> and <var>ram</var> for the two polynomials. The polynomial given by the
coefficients in <var>unram</var> must be a cyclotomic polynomial and the
polynomial given by <var>ram</var> an Eisenstein-polynomial (or 1+x). <strong>This is
not checked by <font face="Gill Sans,Helvetica,Arial">GAP</font>.</strong>
<p>
Every number out of <i>L</i> is represented as a coeff.-list for
the basis {1,<i>x</i>,<i>x</i><sup>2</sup>,…,<i>y</i>,<i>xy</i>,<i>x</i><sup>2</sup><i>y</i>,…} of <i>L</i>. The integer
<var>precision</var> is the number of ``digits'' that all the coefficients have.
<p>
A general comment: the polynomials with which
<code>PadicExtensionNumberFamily</code> is called define an extension of <i>Q</i><sub><i>p</i></sub>. It
must be ensured that both polynomials are really irreducible over
<i>Q</i><sub><i>p</i></sub>! For example x^2+x+1 is <strong>not</strong> irreducible over Q_p. Therefore the
``extension'' PadicExtensionNumberFamily(3, 4, [1,1,1], [1,1]) contains
non-invertible ``pseudo-<i>p</i>-adic numbers''. Conversely, if an
``extension'' contains noninvertible elements one of the polynomials was
not irreducible.
<p>
<a name = "SSEC002.2"></a>
<li><code>PadicNumber( </code><var>fam</var><code>, </code><var>rat</var><code> ) O</code>
<li><code>PadicNumber( </code><var>purefam</var><code>, </code><var>list</var><code> ) O</code>
<li><code>PadicNumber( </code><var>extfam</var><code>, </code><var>list</var><code> ) O</code>
<p>
create a <i>p</i>-adic number in the <i>p</i>-adic numbers family <var>fam</var>. The
first usage returns the <i>p</i>-adic number corresponding to the rational
<var>rat</var>.
<p>
The second usage takes a pure <i>p</i>-adic numbers family <var>purefam</var> and a
list <var>list</var> of length 2 and returns the number <code>p^</code><var>list</var><code>[1] *
</code><var>list</var><code>[2]</code>. It must be guaranteed that no entry of list[2] is
divisible by the prime p. (Otherwise precision will get lost.)
<p>
The third usage creates a number in the family <var>extfam</var> of a <i>p</i>-adic
extension. The second entry must be a list <var>L</var> of length 2 such that
<var>list</var>[2] is the list of coeff. for the basis
{1,…,<i>x</i><sup><i>f</i>−1</sup>·<i>y</i><sup><i>e</i>−1</sup>} of the extended <i>p</i>-adic field and
<var>list</var>[1] is a common <i>p</i>-part of all the coeff.
<p>
<i>p</i>-adic numbers allow the usual field operations.
<p>
<pre>
gap> efam:=PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1]);;
gap> PadicNumber(efam,7/9);
padic(120(3),0(3))
</pre>
<p>
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 <code>[ 0, [ 1.2000, 0.1210
] ]</code> or as <code>[-1, [ 12.000, 1.2100 ] ]</code> (here the coefficients have to be
multiplied by <i>p</i><sup>−1</sup>).
<p>
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 <code>[-3, [ 0.0012, 0.0022 ] ]</code> and so has in fact maximum
precision.
<p>
<a name = "SSEC002.3"></a>
<li><code>IsPadicExtensionNumber( </code><var>obj</var><code> ) C</code>
<p>
<a name = "SSEC002.4"></a>
<li><code>IsPadicExtensionNumberFamily( </code><var>fam</var><code> ) C</code>
<p>
<p>
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<font face="Gill Sans,Helvetica,Arial">GAP 4 manual<br>December 2008
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