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

Checks whether a small factorbasis generates
the same subgroup of the classgroup as a large
factorbasis.

<p><p>

<h3>
Syntax:
</h3>
<pre>r := OrderClassGroupFactorBasisProve(o, lb, ub);</pre>

<p>

<table>
<tr>
<td>boolean</td>
<td><pre>  r  </pre></td>
<td></td>
</tr>
<tr>
<td>order</td>
<td><pre>  o  </pre></td>
<td></td>
</tr>
<tr>
<td>integer</td>
<td><pre>  lb, ub  </pre></td>
<td>lower resp.~upper bound</td>
</tr>
</table>

<p>


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

<p>

<h3>
Description:
</h3>
This function is used to check whether a given small
factorbasis generates the same subgroup of the classgroup as a
large one. It is recommended first to compute the class group
using a small bound, i.~e.~a small factorbasis, and then
afterwards to check if the class group was computed
itself.

Actually this function is not really needed since one
has a good criterion by the euler product but which is not
fully proven. This function is only able to detect the equality
of the factor bases. It should be used only for small fields
and may take a very long time.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Compute the class group structure of {\Bbb Q}(\sqrt[4]{-65}). 
<pre>

kash> O := OrderMaximal(Z,4,-65);;
kash> OrderClassGroup (O, 100, "Euler");
[ 128, [ 2, 4, 4, 4 ] ]
kash> OrderMinkowski(O);
1274
kash> OrderClassGroupFactorBasisProve(O, 100, 1274);
> true

</pre>

<p>


<hr>

<p>
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