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

Returns the number of roots of unity in the given order.

<p><p>

<h3>
Syntax:
</h3>
<pre>r := OrderTorsionUnitRank(o);</pre>

<p>

<table>
<tr>
<td>integer</td>
<td><pre>  r  </pre></td>
<td>number of roots of unity</td>
</tr>
<tr>
<td>order</td>
<td><pre>  o  </pre></td>
<td></td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#OrderTorsionUnit">OrderTorsionUnit</a>, <a href="kashref.html#OrderSetTorsionUnit">OrderSetTorsionUnit</a>

<p>

<h3>
Description:
</h3>
o must be an absolute order. <tt> OrderTorsionUnitRank</tt>
returns the number of roots of unity in the order o,
the rank of the stored generator of the roots of unity in
the order o.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Compute the number of roots of unity in {Z}[\sqrt{-3}]
and in {Z}[frac{1+\sqrt{-3}}{2}] 
<pre>

kash> o := Order(Z,2,-3);
Generating polynomial: x^2 + 3

kash> O := OrderMaximal(o);
   F[1]
    |
   F[2]
  /
 /
Q
F  [ 1]     Given by transformation matrix
F  [ 2]     x^2 + 3
Discriminant: -3 

kash> OrderTorsionUnitRank(o);
2
kash> OrderTorsionUnitRank(O);
> 6

</pre>

<p>


<hr>

<p>
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