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

Calculates a <tt> p</tt>-adic factorization
of the defining polynomial of an order.

<p><p>

<h3>
Syntax:
</h3>
<pre>L := OrderPolyHenselLift(o, d, p, k);</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>
<tr>
<td>integer</td>
<td><pre>  d  </pre></td>
<td></td>
</tr>
<tr>
<td>prime</td>
<td><pre>  p  </pre></td>
<td></td>
</tr>
<tr>
<td>integer</td>
<td><pre>  k  </pre></td>
<td></td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
Let o be an order defined by a polynomial
f \in Z[x] and let p \in Z be a prime
which does not divide the discriminant of f.
This function computes the factorization of f
over the unramified extension of Q|p of
degree d. It returns a list containing
the factors modulo p^{2^{<tt> k</tt>}}.

<p><p>

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

<pre>

kash> o :=Order(Poly(Zx,[1,0,-4,0,1]));
Generating polynomial: x^4 - 4*x^2 + 1

kash> P :=OrderPolyHenselLift(o,2,5,2);
[ x + [258, 312], x + [-258, 312], x + [-258, -312], x + [258, -312] ]
kash> P[1]*P[2]*P[3]*P[4];
x^4 - 522504*x^2 + 16415759376
kash> P :=OrderPolyHenselLift(o,3,7,2);
[ x^2 - 235*x - 1, x^2 + 235*x - 1 ]
kash> P[1]*P[2];
> x^4 - 55227*x^2 + 1

</pre>

<p>


<hr>

<p>
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