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

Computes the minimal polynomial of an algebraic element, a matrix,
or an alff element

<p><p>

<h3>
Syntax:
</h3>
<pre>p := MinPoly (a [,PA]);
p := MinPoly (a [,O]);
p := MinPoly( M );
p := MinPoly(a);</pre>

<p>

<table>
<tr>
<td>polynomial</td>
<td><pre>  p  </pre></td>
<td></td>
</tr>
<tr>
<td>polynomial algebra</td>
<td><pre>  PA  </pre></td>
<td></td>
</tr>
<tr>
<td>suborder</td>
<td><pre>  O  </pre></td>
<td></td>
</tr>
<tr>
<td>algebraic element alff order element</td>
<td><pre>  a  </pre></td>
<td></td>
</tr>
<tr>
<td>matrix</td>
<td><pre>  M  </pre></td>
<td></td>
</tr>
</table>

<p>


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

<p>

<h3>
Description:
</h3>
For an algebraic element a in an Order O and a polynomial
algebra PA over another order o or this order,
this function computes the minimal polynomial of a over o
i.e. the returned polynomial is contained in PA.
Given a matrix this function returns its minimal polynomial.
Given a alff element this function computes the minimal polynomial

<p><p>

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

<pre>

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

kash> a := Elt(o,[0,1]);
[0, 1]
kash> p := MinPoly(a);
x^2 - 2
kash> M := Mat(o,[[1,a],[1,-a]]);
[1 [0, 1]]
[1 [0, -1]]
kash> p := MinPoly(M);
> x^2 + [-1, 1]*x + [0, -2]

</pre>

<p>


<hr>

<p>
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