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

Tests if a class field is also abelian over Q.

<p><p>

<h3>
Syntax:
</h3>
<pre>a:=RayClassFieldIsAbelian(G [,m]);</pre>

<p>

<table>
<tr>
<td>AbelianGroup</td>
<td><pre>  G  </pre></td>
<td>defining a class group</td>
</tr>
<tr>
<td>boolean</td>
<td><pre>  a  </pre></td>
<td></td>
</tr>
<tr>
<td>integer</td>
<td><pre>  m  </pre></td>
<td></td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
Let O be the ring of integers of an abelian extention over Q.
{\goth m} is a multiple of its conductor {\goth f} \in Z.
G is a class group in O, K the abelian extension over O defined
by G. The function determines if K is also abelian over Q.
The methode is described in cite[II, p.24]{HasseZahlbericht}.


<p><p>

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

<pre>

kash> o:=OrderMaximal(x^2-10);
Generating polynomial: x^2 - 10
Discriminant: 40 

kash> G:=RayClassGroupToAbelianGroup(9*o);
RayClassGroupToAbelianGroup(<9>, [  ])
Group with relations:
[2 0]
[0 3]

kash> RayClassFieldAbelianTest(G);
true


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Let's check this. First we'll compute the <tt> RayClassField</tt>
<pre>

kash> L:=RayClassField(G);
[ x^2 - 2, x^3 - 3*x - 1 ]


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
The only critical part is the cubic polynomial, therfore we'll show
that this polynomial defines a C|3 extension over Q.
<pre>

kash> g:=PolyMove(L[2], Z);
x^3 - 3*x - 1
kash> OrderAutomorphismsAbel(Order(g));
true
kash> o:=OrderMaximal(x^4-2);
Generating polynomial: x^4 - 2
Discriminant: -2048 

kash> G:=RayClassGroupToAbelianGroup(9*o);
RayClassGroupToAbelianGroup(<9>, [  ])
Group with relations:
[3 0]
[0 3]
kash> RayClassFieldIsAbelian(G);
false

</pre>

<p>


<hr>

<p>
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