[back] [prev] [next] [index] [root]

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

Computes or stores automorphisms of the given extension.

<p><p>

<h3>
Syntax:
</h3>
<pre>aut := OrderAutomorphisms(o);
aut := OrderAutomorphisms(o, L);
aut := OrderAutomorphisms(o, "normal");
aut := OrderAutomorphisms(o, "abel");</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  aut  </pre></td>
<td>list of automorphisms</td>
</tr>
<tr>
<td>order</td>
<td><pre>  o  </pre></td>
<td>the given order</td>
</tr>
<tr>
<td>list</td>
<td><pre>  L  </pre></td>
<td>list of some known automorphisms</td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#EltAutomorphism">EltAutomorphism</a>, <a href="kashref.html#OrderAutomorphismsAbel">OrderAutomorphismsAbel</a>, <a href="kashref.html#OrderAutomorphismsNormal">OrderAutomorphismsNormal</a>

<p>

<h3>
Description:
</h3>
This function computes or stores the automorphisms of the
given extension. The automorphisms are represented by algebraic
numbers which are zeros of the generating polynomials of the
given extension. They can be applied to algebraic numbers
with the function <tt> EltAutomorphism</tt>.

In the case that some automorphisms are
known in the list <tt> L</tt>, they can be stored with
<tt> OrderAutomorphisms(o, L);</tt>.

In the other cases the automorphisms will be computed. The option
<tt> "normal"</tt> can be omitted. With the option <tt> "abel"</tt> you
can turn on the more efficient algorithm for abelian extensions.
The computation of automorphisms is only possible for absolute
normal extensions. In the case that an extension is not normal
the function will return <tt> false</tt>. The algorithms
are described in Kl2,AcKl1.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Store an automorphism:
<pre>

kash> o := Order(x^4-4*x^2+1);;
kash> a := Elt(o, [0,-1,0,0]);;
kash> OrderAutomorphisms(o, [a]);
[ [0, 1, 0, 0], [0, -1, 0, 0] ]


</pre>

<p>

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

Compute the automorphisms:
<pre>

kash> o := Order(x^4-4*x^2+1);;
kash> OrderAutomorphisms(o);
[ [0, 1, 0, 0], [0, -4, 0, 1], [0, 4, 0, -1], [0, -1, 0, 0] ]


</pre>

<p>

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

Compute the automorphisms in the abelian case (faster):
<pre>

kash> o := Order(x^4-4*x^2+1);;
kash> OrderAutomorphisms(o, "abel");
> [ [0, 1, 0, 0], [0, -4, 0, 1], [0, 4, 0, -1], [0, -1, 0, 0] ]

</pre>

<p>


<hr>

<p>
<a href="../kashref.html"><- back</a>[back] [prev] [next] [index] [root]