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

Enumerates the next prime ideal.

<p><p>

<h3>
Syntax:
</h3>
<pre>p := IdealPrimeCountNext(s [, 1]);</pre>

<p>

<table>
<tr>
<td>prime ideal</td>
<td><pre>  p  </pre></td>
<td></td>
</tr>
<tr>
<td>record</td>
<td><pre>  s  </pre></td>
<td>from IdealPrimeCountInit</td>
</tr>
<tr>
<td>if present, p will be a pair [p, f] with p a prime of the coefficient order and <tt> f</tt> the degree of <tt> p</tt></td>
<td><pre>  1  </pre></td>
<td></td>
</tr>
</table>

<p>


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

<p>

<h3>
Description:
</h3>
Be careful with relative extensions: index divisors won't work here!

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
We'll work in the field Q(\sqrt 2):
<pre>

kash> o := OrderMaximal(Z, 2, 2);
Generating polynomial: x^2 - 2
Discriminant: 8 

kash> s := IdealPrimeCountInit(o);
Record of type IdealPrimeCount

kash> repeat
> p := IdealPrimeCountNext(s);
> Print(p, "\n");
> until IdealMin(p)>20;
<2, [0, 1]>
<3>
<5>
<7, [3, 1]>
<7, [4, 1]>
<11>
<13>
<17, [6, 1]>
<17, [11, 1]>
<19>
<23, [5, 1]>


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Next, only the factorization shape of ideals coprime to 2 is
investigated:
<pre>

kash> s := IdealPrimeCountInit(o, 2);
Record of type IdealPrimeCount

kash> repeat
> fs := IdealPrimeCountNext(s, 1);
> Print(fs, "\n");
> until fs[1]>20;
> [ 3, 2 ]
[ 5, 2 ]
[ 7, 1 ]
[ 11, 2 ]
[ 13, 2 ]
[ 17, 1 ]
[ 19, 2 ]
[ 23, 1 ]

</pre>

<p>


<hr>

<p>
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