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

Determines the Hilbert class field of an imaginary
quadratic field.

<p><p>

<h3>
Syntax:
</h3>
<pre>O := ImQuadHilbert(o [,repr] [,"roots"] );</pre>

<p>

<table>
<tr>
<td>order</td>
<td><pre>  o  </pre></td>
<td>imaginary quadratic field</td>
</tr>
<tr>
<td>list</td>
<td><pre>  repr  </pre></td>
<td>representants for the ideal classes of o</td>
</tr>
<tr>
<td>order</td>
<td><pre>  O  </pre></td>
<td>Hilbert class field</td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#OrderHilbert">OrderHilbert</a>, <a href="kashref.html#ImQuadRayField">ImQuadRayField</a>, <a href="kashref.html#RayClassField">RayClassField</a>

<p>

<h3>
Description:
</h3>
Let K be an imaginary quadratic field and K|{(1)} its Hilbert
class field.
Possible generators for K|{(1)} over K are for example
quotients of the Dedekind \eta-function of the form
 \left(frac{\eta(a)\eta(b)}{\eta(ab)\eta(O|K)}right)^n,quad n\in\N, 
where the integral O|K-ideals a and b have
to be suitably normalized (this is implicitly contained
in Scherz)
A defining equation for O|{K|{(1)}} is calculated
using complex
approximations one gets by evaluating the defining infinite
series of the Dedekind \eta-function.

If <tt> "roots"</tt> is specified, also the roots of the minimal polynomial
in the ordering given by the ordering of the used ideal class
representants (that correspond to the Galois automorphisms
by use of the Artin map) are returned.

<p><p>

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

<pre>

</pre>

<p>

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

Test for correctness (in other examples the coefficients of
the generating polynomial one gets by using the routine
<tt> ImQuadHilbert</tt> are usually quite small):
<pre>

</pre>

<p>


<hr>

<p>
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