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

Returns a principal generator of an ideal, if it is a
principal ideal and <tt> false</tt> otherwise.

<p><p>

<h3>
Syntax:
</h3>
<pre>g := IdealIsPrincipal(I);
g := IdealIsPrincipal(I, classgroup);</pre>

<p>

<table>
<tr>
<td>ideal </td>
<td><pre>  I  </pre></td>
<td></td>
</tr>
<tr>
<td>false or algebraic number</td>
<td><pre>  g  </pre></td>
<td>principal generator</td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
There are two different algorithms used by this function. The
standard procedure is not an easy check and can include
expensive enumerations. If a second argument "classgroup" is
given, it uses precomputed data from class group
calculations. In this case it is often quite fast. Once
computed, the result is stored and the ideal arithmetic uses
the principal generator. A warning: The standard procedure for
number fields
uses real arithmetic and the correctness of the results depends
on the precision defined by <tt> OrderPrec()</tt> whereas the
second method works unconditional.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
We will test the prime ideal lying over 2 in the maximal order
of x^3 + 6x^2 + 6x + 6.
<pre>

kash> O := OrderMaximal(Order(Poly(Zx, [1, 6, 6, 6])));
Generating polynomial: x^3 + 6*x^2 + 6*x + 6
Discriminant: -1836 

kash> p := Factor(O, 2*O)[1][1];
<2, [0, 1, 0]>
kash> IdealIsPrincipal(p);
false
kash> IdealIsPrincipal(p^3);
> 2

</pre>

<p>


<hr>

<p>
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