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

Computes the relative units.

<p><p>

<h3>
Syntax:
</h3>
<pre>L := OrderRelUnits(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#OrderSUnits">OrderSUnits</a>

<p>

<h3>
Description:
</h3>
The <tt> OrderRelUnits</tt> function returns a basis of the
S-units modulo split into torsion units and other S-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 ideal 1*o.

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

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
We have for example:
<pre>

kash> o:= Order(Poly(Zx,[1,1,1,1,1]));
Generating polynomial: x^4 + x^3 + x^2 + x + 1

kash> zeta:= Elt(o,[0,1,0,0]);
[0, 1, 0, 0]
kash> ox:=PolyAlg(o);
Univariate Polynomial Ring in x over Generating polynomial: x^4 + x^3 + x^2 + \
x + 1

kash> O:= Order(Poly(ox,[1,0,-zeta,0,zeta^2]));
      F[1]
        /
       /
   E1[1]
  /
 /
Q
F  [ 1]     x^4 + [0, -1, 0, 0]*x^2 + [0, 0, 1, 0]
E 1[ 1]     x^4 + x^3 + x^2 + x + 1

kash> I:=31*o;
<31>
kash> m:=OrderRelUnits(O,I);
> [ [ [0, [-2, -2, -1, -1], 0, [0, 1, 0, -1]], 
      [0, [-1, -1, -1, 0], 0, [-1, -1, -2, -1]], 
      [[0, -1, 0, 0], [0, 0, -1, -1], [1, 1, 1, 0], [0, 1, 1, 1]], 
      [-1, [0, 0, 1, 0], [0, 0, -1, 0], [0, -1, 0, -1]], 
      [[-1, -1, -1, -1], [0, 1, 0, 0], [0, 0, 0, -1], -1], 
      [[-1, 0, -1, 0], [1, 0, 0, 1], [-1, -1, -1, -1], [0, 0, -1, 0]], 
      [[-10, -19, -24, -44], 0, [-14, -8, 16, -12], 0] / 31, 
      [[17, 10, 22, -3], 0, [-4, -37, -7, -23], 0] / 31, 
      [[-66, -67, -137, -77], 0, [6, -39, -58, -117], 0] / 31, 
      [[-19, -26, 84, 56], 0, [27, -43, -59, -91], 0] / 31 ], 
  [ [[0, 0, 1, 1], 0, [-1, -1, -1, 0], 0], 
      [0, [0, -1, -1, 1], 0, [2, 1, 1, 1]], 
      [[-49, -38, -43, -52], 0, [27, -5, 18, -15], 0] / 31, 
      [[-2, -18, 71, 8], 0, [34, -4, 33, 81], 0] / 31, 
      [[-13, 69, -19, -103], 0, [66, 67, 137, -16], 0] / 31 ] ]

</pre>

<p>


<hr>

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