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

Returns an element with certain valuations at prime ideals.

<p><p>

<h3>
Syntax:
</h3>
<pre>E := EltApproximation(P, L);</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  P  </pre></td>
<td>of distinct prime ideals over the same order </td>
</tr>
<tr>
<td>list</td>
<td><pre>  L  </pre></td>
<td>of small integers </td>
</tr>
<tr>
<td>algebraic element</td>
<td><pre>  E  </pre></td>
<td></td>
</tr>
</table>

<p>


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

<p>

<h3>
Description:
</h3>
This is a version of the approximation theorem for algebraic
numbers over number rings.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Weak approximation theorem in {\Bbb Q}(rho) with rho^3 - 12rho + 10 = 0. 
<pre>

kash> k := OrderMaximal (Order (x^3 - 12*x + 10));
Generating polynomial: x^3 - 12*x + 10
Discriminant: 4212 

kash> P := Filtered (Flat (Factor (56*k)), IsIdeal);
[ <2, [0, 1, 0]>, <7, [1, 1, 0]>, <7, [3, 6, 1]> ]
kash> L := [3,6,7];
[ 3, 6, 7 ]
kash> mu := EltApproximation (P,L);
[10117814, -4470662, -470596]
kash> for I in P do Print (I," -> ", Valuation (I, mu),"\n"); od;
> <2, [0, 1, 0]> -> 3
<7, [1, 1, 0]> -> 6
<7, [3, 6, 1]> -> 7

</pre>

<p>


<hr>

<p>
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