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

Returns the image of an algebraic element under the Minkowski map.

<p><p>

<h3>
Syntax:
</h3>
<pre>v := EltMinkowski(a);</pre>

<p>

<table>
<tr>
<td>real matrix</td>
<td><pre>  v  </pre></td>
<td></td>
</tr>
<tr>
<td>algebraic element</td>
<td><pre>  a  </pre></td>
<td></td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
Let F = Q(O) be the field of fractions of the order O,
which contains the algebraic element a and let
\sigma|1, &hellip; ,\sigma|n be
the Galois isomorphisms of F, where \sigma|1, &hellip; ,\sigma|{r|1}
are precisely the real isomorphisms and overline{\sigma|{j+r|2}} =
\sigma|{j} for r|1 < j  &lt;=  r|1+r|2 are the complex ones. Then
the image of a under the
Minkoswski map \psi : O \mapsto {R}^{r|1+2r|2} is defined as
\begin{eqnarray*}
& [\sigma|1(a), &hellip; ,\sigma|{r|1}(a),
\sqrt2Re(\sigma|{r|1 + 1}(a)),
quad  &hellip; ,\sqrt2Re(\sigma|{r|1+r|2}(a)),

& \sqrt2\Im (\sigma|{r|1 + 1}(a)),quad  &hellip; ,
\sqrt2\Im (\sigma|{r|1+r|2}(a))].&
\end{eqnarray*}

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Matrix of logarithms of archimedean valuations
of an algebraic element. 
<pre>

kash> O := Order (Z,2,-1);
Generating polynomial: x^2 + 1

kash> rho := Elt (O,[0,1]);
[0, 1]
kash> EltMinkowski (rho);
[ 0, 1.414213562373095048801688724209698078569671875377]
kash> O := Order (Poly(Zx,[1,0,73,-280,-2399]));
Generating polynomial: x^4 + 73*x^2 - 280*x - 2399

kash> rho := Elt (O,[1,1,0,1]);
[1, 1, 0, 1]
kash> EltMinkowski (rho);
> [ 234.532336541434390915922266933690607679916764155403, -60.735882043911047217\
34054416286226981382540207225, 473.9054718000681082230949125242621731996360759\
3719, -1354.559212605075329197656456425992015910014499693796]

</pre>

<p>


<hr>

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