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

Computes the S-unit group.

<p><p>

<h3>
Syntax:
</h3>
<pre>L := OrderSUnits(o [,S  |  I] [,"raw"]);</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  L  </pre></td>
<td>of L1, T, hs or L2, hs</td>
</tr>
<tr>
<td>list</td>
<td><pre>  L1  </pre></td>
<td>of algebraic integers</td>
</tr>
<tr>
<td>list</td>
<td><pre>  L2  </pre></td>
<td>of S-units</td>
</tr>
<tr>
<td>matrix</td>
<td><pre>  T  </pre></td>
<td>transformation matrix</td>
</tr>
<tr>
<td>integer</td>
<td><pre>  hs  </pre></td>
<td>S-class number</td>
</tr>
<tr>
<td>order</td>
<td><pre>  o  </pre></td>
<td></td>
</tr>
<tr>
<td>list</td>
<td><pre>  S  </pre></td>
<td>of pairwise distinct prime ideals.</td>
</tr>
<tr>
<td>ideal</td>
<td><pre>  I  </pre></td>
<td></td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#OrderClassGroup">OrderClassGroup</a>, <a href="kashref.html#OrderClassGroupCheck">OrderClassGroupCheck</a>, <a href="kashref.html#OrderSUnitsPositive">OrderSUnitsPositive</a>

<p>

<h3>
Description:
</h3>
The <tt> OrderSUnits</tt> function returns a basis of the
S-units modulo torsion units. You may specify S either
as a list of prime ideals or as an ideal {\goth{I}}. In this case
S consists of all primes dividing {\goth{I}}. Omitting the
argument S means to choose for S the factor basis used
during class group computation. The first elements of this
basis are fundamental units.

If the last argument is <tt> "raw"</tt>,
this function will return some algebraic integers \alpha|1,
\dots, \alpha|s and a transformation matrix T such that
 (\alpha|1, \dots, \alpha|s)T  interpreted as power
products gives a basis of the S-unit group modulo torsion
units.

In both cases it will also
return the cardinality of the subgroup of
the class group which is generated by the prime ideals of
S.

The first basis elements are not fully proven
to be fundamental units. This may be verified using
<tt> OrderClassGroupCheck</tt>.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
For Q(\sqrt[3]{333}) we have for example:
<pre>

kash> o := OrderMaximal(Z, 3, 333);;
kash> OrderClassGroup(o, 500, euler, fast);
[ 3, [ 3 ] ]
kash> OrderSUnits(o, 7*11*13*o);
> [ [ [-7309383190, 288532187, 331545960], -7, [256, 37, 16], [-13, -12, 6], 
      [3422, -161, -144], [4852, 700, 303], [16, 0, -1] ], 3 ]

</pre>

<p>


<hr>

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