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

Finds the maximal factorgroup where a list of automorphisms
acts trivial.

<p><p>

<h3>
Syntax:
</h3>
<pre>sg := FindMaximalCentralField(o | I [, inf][, aut]);
sg := FindMaximalCentralField(G [, aut]);</pre>

<p>

<table>
<tr>
<td>Matrix</td>
<td><pre>  sg  </pre></td>
<td>describing the aditional relations</td>
</tr>
<tr>
<td>order</td>
<td><pre>  o  </pre></td>
<td></td>
</tr>
<tr>
<td>integral ideal</td>
<td><pre>  I  </pre></td>
<td></td>
</tr>
<tr>
<td>list</td>
<td><pre>  inf  </pre></td>
<td>of infinite places</td>
</tr>
<tr>
<td>AbelianGroup</td>
<td><pre>  G  </pre></td>
<td>must be from RayClassGroupToAbelianGroup</td>
</tr>
<tr>
<td>list</td>
<td><pre>  aut  </pre></td>
<td>of automorphisms of <tt> o</tt>, if omitted  <tt> OrderAutomorphisms(o, [])</tt> is used.</td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
<i>no detailed description available yet</i>

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
We'll investigate the ray class field mod m := (6)\p|1^\infty\p|2^\infty in k := Q(\sqrt{10}):
<pre>

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

kash> OrderAutomorphisms(o);
[ [0, 1], [0, -1] ]
kash> G := RayClassGroupToAbelianGroup(6*o, [1, 2]);
RayClassGroupToAbelianGroup(<6>, [ 1, 2 ])
Group with relations:
[2 0 0]
[0 4 0]
[0 0 2]
kash> rcg := FindMaximalCentralField(G);
[2 0 0]
[0 2 0]
[0 0 2]
[0 0 0]
[0 0 0]
[0 0 0]


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Since <tt> rcg</tt> corresponds not to the whole group, we can construct two fields
that are normal over Q: the ray class field modulo m and the maximal central
extension of k:
<pre>

kash> cf1 := RayClassField(6*o, [1, 2], rcg);
[ x^2 - 6, x^2 - 2, x^2 + 2 ]
kash> cf2 := RayClassField(6*o, [1, 2]);
[ x^2 - 6, x^2 + 2, x^4 + [8, -4]*x^2 + [28, -8] ]


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Finally we'll compute defining equations and the automorphisms:
<pre>

kash> O1 := RayClassFieldAuto(cf1);
[       F[1]
            /
           /
       E1[1]
      /
     /
    Q
    F  [ 1]     x^8 - 24*x^6 + 248*x^4 + 288*x^2 + 2704
    E 1[ 1]     x^2 - 10
    Discriminant: <17352869066524232277088370178392064> 
    , [ a, a*b, a*b*c, a*b*c*d, a*b*d, a*c, a*c*d, a*d, b, b*c, b*c*d, b*d, 
      c, c*d, d, e ] ]


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
 &hellip;  and see that <tt> O1</tt> is abelian over Q whereas <tt> O2</tt> is non-abelian. Since
k is cyclic over Q, central extensions coincide with absolute abelian ones.
<pre>

kash> a := O1[2][Position(O1[2], "a")];
a
kash> b := O1[2][Position(O1[2], "b")];;
kash> c := O1[2][Position(O1[2], "c")];;
kash> d := O1[2][Position(O1[2], "d")];;
kash> a*b = b*a; a*c = c*a; a*d = d*a;
true
true
true
kash> b*c = c*b; b*d = d*b; c*d = d*c;true
true
> true

</pre>

<p>


<hr>

<p>
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