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

Minimal polynomial of an algebraic element over a subfield.

<p><p>

<h3>
Syntax:
</h3>
<pre>p := EltMinPoly (a [, PA]);
p := EltMinPoly (a [, O]);</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</td>
<td><pre>  a  </pre></td>
<td></td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
Given 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.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Minimal polynomials of elements in
{\Bbb Q}(\sqrt{10},\sqrt{5}, \sqrt{3}). 
<pre>

kash> E1 := Order (Z,2,10);
Generating polynomial: x^2 - 10

kash> E2 := Order (E1, 2, 5);
      F[1]
        /
       /
   E1[1]
  /
 /
Q
F  [ 1]     x^2 - 5
E 1[ 1]     x^2 - 10

kash> F := Order (E2, 2, 3);
         F[1]
           /
          /
      E2[1]
        /
       /
   E1[1]
  /
 /
Q
F  [ 1]     x^2 - 3
E 2[ 1]     x^2 - 5
E 1[ 1]     x^2 - 10

kash> E1x := PolyAlg (E1);
Univariate Polynomial Ring in x over Generating polynomial: x^2 - 10

kash> Fx := PolyAlg (F);
Univariate Polynomial Ring in x over          F[1]
           /
          /
      E2[1]
        /
       /
   E1[1]
  /
 /
Q
F  [ 1]     x^2 - 3
E 2[ 1]     x^2 - 5
E 1[ 1]     x^2 - 10

kash> a := Elt (F, [[[1,2],[3,4]],[[3/5, 5],[7,9]]]);
[[[5, 10], [15, 20]], [[3, 25], [35, 45]]] / 5
kash> g := EltMinPoly (a, E1x);
x^4 + [-4, -8]*x^3 + [-717904, -100800] / 25*x^2 + [-3443792, -2723584] / 25*x\
 + [99481555504, 24875390400] / 625
kash> f := PolyMove (g, Fx);
x^4 + [[[-4, -8], 0], 0]*x^3 + [[[-717904, -100800], 0], 0] / 25*x^2 + [[[-344\
3792, -2723584], 0], 0] / 25*x + [[[99481555504, 24875390400], 0], 0] / 625
kash> Eval (f, a);
> 0

</pre>

<p>


<hr>

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