[back] [prev] [next] [index] [root]

<p>&nbsp;<p>
<hr noshade size="5">
<h2><a name="RayResidueRing"></a>
RayResidueRing
</h2>

This function computes the multiplicative group of the residue
class ring.

<p><p>

<h3>
Syntax:
</h3>
<pre>L := RayResidueRing(m0 [,minf]);</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  L  </pre></td>
<td></td>
</tr>
<tr>
<td>ideal </td>
<td><pre>  m0  </pre></td>
<td></td>
</tr>
<tr>
<td>list</td>
<td><pre>  minf  </pre></td>
<td>of integers/infinite primes</td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#EltCon">EltCon</a>, <a href="kashref.html#EltRayResidueRingRep">EltRayResidueRingRep</a>, <a href="kashref.html#RayResidueRingRepToElt">RayResidueRingRepToElt</a>, <a href="kashref.html#RayResidueRingCyclicFactors">RayResidueRingCyclicFactors</a>

<p>

<h3>
Description:
</h3>
This function computes the multiplicative group of the residue
class ring of
an ideal or a congruence module
{\goth{m}} = {\goth{m}}_0{\goth{m}}_\infty, where
{\goth{m}}_0
is an integral ideal of a maximal
order O and {\goth{m}}_\infty is a formal product of primes
at infinity.
{\goth{m}}_\infty is represented by a list of integers, where
each entry represents a real embedding (as given by {\tt
EltCon}).
<tt> RayResidueRing</tt> returns a list of the order of the
multiplicative group and a list
of the orders of the cyclic factors.

This structure can be used to solve multiplicative
congruences which may involve infinite places. This algorithm
bases an the presentation of the Einseinheiten in Hablau
and is described in Pau1.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>

<pre>

kash> O := OrderMaximal(Order(Z,2,10));
Generating polynomial: x^2 - 10
Discriminant: 40 

kash> P3 := Factor(3*O)[1][1];;
kash> IdealDegree(P3);
1
kash> IdealRamIndex(P3);
1
kash> m0 := P3^2;; L := RayResidueRing(m0);
[ 6, [ 2, 3 ] ]
kash> m0 := P3^4;; L := RayResidueRing(m0);
[ 54, [ 2, 27 ] ]
kash> P5 := Factor(5*O)[1][1];;
kash> IdealDegree(P5);
1
kash> IdealRamIndex(P5);
2
kash> m0 := P5^2;; L := RayResidueRing(m0);
[ 20, [ 4, 5 ] ]
kash> m0 := P5^4;; L := RayResidueRing(m0);
[ 500, [ 4, 25, 5 ] ]
kash> P7 := 7*O;;
kash> IdealDegree(P7);
2
kash> IdealRamIndex(P7);
1
kash> m0 := P7^2;; RayResidueRing(m0);
[ 2352, [ 48, 7, 7 ] ]
kash> m0 := P7^4;; RayResidueRing(m0);
[ 5647152, [ 48, 343, 343 ] ]
kash> m0 := P5^4*P7^2;; L := RayResidueRing(m0);
[ 1176000, [ 4, 25, 5, 48, 7, 7 ] ]
kash> minf := [2];; L := RayResidueRing(1*O,minf);
[ 2, [ 1, 2 ] ]
kash> L := RayResidueRing(m0,minf);
[ 2352000, [ 4, 25, 5, 48, 7, 7, 2 ] ]


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
 compute the ray residue ring of an ideal in Z 
<pre>

kash> a := ZIdealCreate(9);
<9>
kash> RayResidueRing(a);
[ 6, [ 6 ] ]

</pre>

<p>


<hr>

<p>
<a href="../kashref.html"><- back</a>[back] [prev] [next] [index] [root]