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

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

Computes a maximal exponent.

<p><p>

<h3>
Syntax:
</h3>
<pre>L := OrderKextModularPower(pI, mu);</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  L  </pre></td>
<td>containing alpha and l</td>
</tr>
<tr>
<td>ideal</td>
<td><pre>  pI  </pre></td>
<td>a prime ideal</td>
</tr>
<tr>
<td>algebraic element</td>
<td><pre>  mu  </pre></td>
<td></td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
Given an prime ideal \p\subset o and an integral element
\mu\ino\setminus\p, such that zeta|p\ino for the prime
number p dividing \p.
This function will compute the maximal exponent
l &lt;=  pfrac {v|{\p\;}(p)}{p-1} such that
there is an \alpha subject to \alpha^p\equiv\mu \pmod {\p^l}.
\alpha and l will be returned.

This function is useful mainly in the context of Kummer extensions,
where conditions like the above are used to compute the discriminant
and the splitting of \p. This is based on cite[p 50]{Dabe2}.

<p><p>

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

<pre>

kash> o := OrderCyclotomic(7);;
kash> pI := Factor(7*o)[1][1];
<7, [6, 1, 0, 0, 0, 0]>
kash> mu := Elt(o, [1,2,3,4,5,8]);
[1, 2, 3, 4, 5, 8]
kash> L := OrderKextModularPower(pI, mu);
[ 2, 1 ]
kash> EltIsInIdeal(mu - L[1]^7, pI^L[2]);
> true

</pre>

<p>


<hr>

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