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

Solves a (relative) norm equation.

<p><p>

<h3>
Syntax:
</h3>
<pre>L := OrderNormEquation(o, a [,n | "all" [,"exact" | "abs" | "ineq"]]);</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  L  </pre></td>
<td></td>
</tr>
<tr>
<td>int  |  element of the coefficient ring</td>
<td><pre>  a  </pre></td>
<td></td>
</tr>
<tr>
<td>int</td>
<td><pre>  n  </pre></td>
<td></td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#Solve">Solve</a>

<p>

<h3>
Description:
</h3>
Searches for elements \alpha \in o with norm a.
The third argument specifies the number of different non
associated solutions. <tt> OrderNormEquation</tt> tries to find
n, all or, if unspecified, 1 solution.

The default for the fourth argument is <tt> "exact":</tt>
the function solves the above mentioned equation.
If <tt> "abs"</tt> is given, the norm equation
 rm{Norm}(\alpha) = \epsilon a  is solved, where \epsilon is
a torsion unit of the coefficient ring of the given order.
The <tt> "ineq"</tt>-Option is supported only in the absolute
case, here the norm equation 0  &lt;=  rm{Norm}(\alpha)  &lt;=  a
is solved.

Remember that solving relative norm equations may take
quite a long time. Due to the used algorithm it is
faster to call the <tt> OrderNormEquation</tt> function
using the <tt> "abs"</tt>-Option than using the
<tt> "exact"</tt>-Option or the default value.

For further information about the algorithms we
refer the reader to Fin1 for the absolute case
and to Fi1 otherwise.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
First we solve an absolute norm equation in Z[\sqrt[3]{5}]. 
<pre>

kash> o := Order(Z, 3, 5);
Generating polynomial: x^3 - 5

kash> L := OrderNormEquation(o, 4);
[ [29, 17, 10] ]
kash> L := OrderNormEquation(o, 4, "all");
[ [29, 17, 10], [-1, -1, 1] ]
kash> EltNorm(L[1]);
4
kash> EltNorm(L[2]); 
4

</pre>

<p>

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


Now a relative norm equation is solved in the
maximal order of Q(\sqrt[3]{5},\sqrt{2}). 
<pre>

kash> O := Order(o, 2, 2);;
kash> EltNorm(Elt(O, [[1, 2, 3], [3, 2, 2]]));
[-37, -15, -22]
kash> a:=OrderNormEquation(O, Elt(o, [-37, -15, -22]), 1, "abs");
[ [[-5, -2, -1], [2, 0, -1]] ]
kash> EltNorm(a[1]);
> [37, 15, 22]

</pre>

<p>


<hr>

<p>
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