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

Creates an absolute extension from a simple relative extension
or creates a simple relative extension from a double relative
extension.

<p><p>

<h3>
Syntax:
</h3>
<pre>Oa := OrderAbs(O);
Oa := OrderAbs(O,"no hom");</pre>

<p>

<table>
<tr>
<td>order</td>
<td><pre>  Oa  </pre></td>
<td></td>
</tr>
<tr>
<td>order</td>
<td><pre>  O  </pre></td>
<td></td>
</tr>
</table>

<p>


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

<p>

<h3>
Description:
</h3>
Given a simple relative order O - the coefficient ring
(<tt> OrderCoefOrder</tt>) of the coefficient ring of
O is {\Bbb Z} -
this function computes an absolute equation order O|a
with quotient field isomorphic to the quotient field of
O over Q.

Given a double relative order O - the coefficient ring
of the coefficient ring of O,
denoted by o, has the coefficient ring {\Bbb Z} -
this function computes a relative equation order O|a
with quotient field isomorphic to the quotient field of O over
the quotient field of o. o has to be maximal.

Usually the homomorphisms are computed and installed
(cf. <tt> OrderInstallHom</tt>), so that elements can be moved between
the orders O and O|a.
But this may take a while.
If you want to omit the computation of the homomorphism,
use the second optional argument <tt> "no hom"</tt>.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
The following function merges two absolute orders 
<pre>

kash> Merge := function(o1,o2)
> local o1x, f, fl;
> o1x := PolyAlg(o1);
> f := PolyMove(OrderPoly(Zx, o2), o1x);
> fl := Factor (f);
> return OrderAbs(Order(fl[Length(fl)][1]));
> end;
function ( o1, o2 ) ... end


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
 We create Q(\sqrt5, \sqrt3):

<pre>

kash> o := Order(Z, 2, 3);
Generating polynomial: x^2 - 3

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

kash> Oa := OrderAbs(O);
Generating polynomial: x^4 - 16*x^2 + 4


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
 The element \sqrt3 of o is represented in O|a:

<pre>

kash> a:=EltMove(Elt(o, [0,1]), Oa);
[0, -14, 0, 1] / 4
kash> EltMinPoly(a, Zx);
x^2 - 3


</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
 A double relative extension:

<pre>

kash> o := OrderMaximal(Z,8,2);
Generating polynomial: x^8 - 2
Discriminant: -2147483648 

kash> oo := Order(o,2,5);
      F[1]
        /
       /
   E1[1]
  /
 /
Q
F  [ 1]     x^2 - 5
E 1[ 1]     x^8 - 2

kash> O := Order(oo,2,7);
         F[1]
           /
          /
      E2[1]
        /
       /
   E1[1]
  /
 /
Q
F  [ 1]     x^2 - 7
E 2[ 1]     x^2 - 5
E 1[ 1]     x^8 - 2

kash> Oa := OrderAbs(O);
      F[1]
        /
       /
   E1[1]
  /
 /
Q
F  [ 1]     x^4 - 24*x^2 + 4
E 1[ 1]     x^8 - 2

kash> Oaa:= OrderAbs(Oa);
> Generating polynomial: x^32 - 192*x^30 + 16160*x^28 - 779520*x^26 + 23611832*x\
^24 - 461456256*x^22 + 5726131264*x^20 - 42165517824*x^18 + 155056995992*x^16 \
- 167363594496*x^14 + 79016151424*x^12 + 17149424640*x^10 + 206341230816*x^8 -\
 1685897697792*x^6 - 781759831296*x^4 - 194736998400*x^2 + 415926965776


</pre>

<p>


<hr>

<p>
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