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

Returns the factorization of the given ideal over a maximal order.

<p><p>

<h3>
Syntax:
</h3>
<pre>F := IdealFactor(a);</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  F  </pre></td>
<td></td>
</tr>
<tr>
<td>ideal</td>
<td><pre>  a  </pre></td>
<td></td>
</tr>
</table>

<p>


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

<p>

<h3>
Description:
</h3>
Because a maximal order O is a Dedekind ring every
integral O-ideal has a unique factorization in prime
ideals. More generally, it is possible to factor ideals
in prime ideals over arbitrary orders above the equation
order at least in case they do not divide the discriminant
of the equation order.

The factorization can simply be extended to
fractional ideals: if d is the denominator of \a and \b, the
integral ideal d\a, the ideal \b and the ideal dO
is factorized. The exponents of the factorization of dO
are negated and multiplied to the factorization of \b
simplifying exponents of identical prime ideals.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Factorization of the principal ideal (3) in the maximal order of
{\Bbb Q}(\sqrt{-110}):
<pre>

kash> o := Order(Z,2,-110);
Generating polynomial: x^2 + 110

kash> O := OrderMaximal(o);
Generating polynomial: x^2 + 110
Discriminant: -440 

kash> IdealFactor(3*O);
[ [ <3, [1, 1]>, 1 ], [ <3, [2, 1]>, 1 ] ]
kash> a:=Ideal(O,3);
<3>
kash> IdealFactor(a);
> [ [ <3, [1, 1]>, 1 ], [ <3, [2, 1]>, 1 ] ]

</pre>

<p>


<hr>

<p>
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