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

Returns the finite field defined by a prime ideal.

<p><p>

<h3>
Syntax:
</h3>
<pre>K := IdealResidueField(p);</pre>

<p>

<table>
<tr>
<td>ideal </td>
<td><pre>  p  </pre></td>
<td></td>
</tr>
<tr>
<td>finite field </td>
<td><pre>  K  </pre></td>
<td></td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#IdealResidueFieldIsomorphism">IdealResidueFieldIsomorphism</a>, <a href="kashref.html#EltToFFE">EltToFFE</a>, <a href="kashref.html#FFEToElt">FFEToElt</a>, <a href="kashref.html#RayResidueRing">RayResidueRing</a>

<p>

<h3>
Description:
</h3>
The field can be written as

O / {\goth{p}},

where {\goth{p}} is the prime ideal and O the defining order.

<p><p>

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

<pre>

kash> O := OrderMaximal(Order(Poly(Zx,[1,4,1,-4,-3,7])));
Generating polynomial: x^5 + 4*x^4 + x^3 - 4*x^2 - 3*x + 7
Discriminant: 28442269 

kash> p := Factor(7*O)[1][1];
<7, [0, 1, 0, 0, 0]>
kash> IdealResidueField(p);
Finite field of size 7
kash> a := Elt (O, [1,234,54,57,4]);
[1, 234, 54, 57, 4]
kash> EltToFFE (a, p);
1
kash> a := Elt (O, [4,234,54,57,4]);
[4, 234, 54, 57, 4]
kash> b:=EltToFFE (a, p);
4
kash> TYPE (b);
> "KANT finite field elt"

</pre>

<p>


<hr>

<p>
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