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

Returns a list containing information about the
transformation from a suborder of the given order to
the given order.

<p><p>

<h3>
Syntax:
</h3>
<pre>L := OrderTransformationMatrix(O);</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  L  </pre></td>
<td></td>
</tr>
<tr>
<td>order</td>
<td><pre>  O  </pre></td>
<td></td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#OrderBasis">OrderBasis</a>, <a href="kashref.html#OrderCoefIdeals">OrderCoefIdeals</a>, <a href="kashref.html#Order">Order</a>

<p>

<h3>
Description:
</h3>
In the absolute case the list L contains the denominator d
and the n \times n-matrix T with

(omega|1, &hellip; ,omega|n) = frac{1}{d} (a|1, &hellip; ,a|n)T,

where omega|1, &hellip; ,omega|n is a basis of the given order O
and a|1, &hellip; ,a|n is a basis of a suborder of the given order.
If the order O is the equation order, simply d = 1 and
T = I|n are returned.

In the relative case the list L contains a list of ideals
(1 or n, where n is the relative degree of O), called
the coefficient ideals, and an n \times n-matrix T.

If we get only one coefficient ideal \a, the pseudo basis
of the
order O is o omega|1, &hellip; ,o omega|{n-1}, \a omega|n,
where

(omega|1, &hellip; ,omega|n) = (a|1, &hellip; ,a|n)T,

a|1, &hellip; ,a|n are the basis elements of the pseudo basis
of the suborder of O
and o is the coefficient order of O. In this
case the relative order is called "free" (\a = o)
or "almost free" (\a \neq o).

If we get n coefficient ideals \a|1, &hellip; ,\a|n, the
pseudo basis of the order O is
\a|1 omega|1, &hellip; ,\a|n omega|n, with
omega|1, &hellip; ,omega|n as above.

A relative equation order O is "free", hence
<tt> OrderTransformationMatrix(O)</tt> returns
the ideal o (the coefficient order) and the
n \times n identity matrix.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
An absolute extension: 
<pre>

kash> o := Order(Z,2,165);
Generating polynomial: x^2 - 165

kash> O := OrderMaximal(o);
   F[1]
    |
   F[2]
  /
 /
Q
F  [ 1]     Given by transformation matrix
F  [ 2]     x^2 - 165
Discriminant: 165 

kash> OrderTransformationMatrix(O);
> [ 2, [2 1]
    [0 1] ]

</pre>

<p>


<hr>

<p>
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