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

Returns the decomposition of an unit with respect to
the computed system of fundamental units of a maximal order.

<p><p>

<h3>
Syntax:
</h3>
<pre>F := EltUnitDecompose(u);
L := EltUnitDecompose(u,"expons");</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  F  </pre></td>
<td></td>
</tr>
<tr>
<td>list</td>
<td><pre>  L  </pre></td>
<td></td>
</tr>
<tr>
<td>algebraic element</td>
<td><pre>  u  </pre></td>
<td></td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
Let o be an order over Z. Let \varepsilon_1, &hellip; ,
\varepsilon_r be the system of fundamental units of o
computed by the <tt> OrderUnitsFund</tt> function and let
zeta \in o be the generator of the cyclic torsion subgroup
which is accessible by the <tt> OrderTorsionUnit</tt> function.
Each unit u \in o is of a form
 u = zeta^{m_0} \varepsilon^{m_1} cdots \varepsilon^{m_1} 
with rational integers m_0,m_1, &hellip; ,m_r.\bigskip

<tt> F := EltUnitDecompose(u)</tt>
returns the above decomposition of u. \medskip

<tt> L := EltUnitDecompose(u,"expons")</tt>
returns the exponents m_0,m_1, &hellip; ,m_r of the
above decomposition of u.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Decompose certain units in {Z}[\sqrt[4]{2}] :
<pre>

kash> o := Order(Z,4,2);
Generating polynomial: x^4 - 2

kash> L := OrderUnitsFund(o);
[ [1, 0, 1, 0], [1, -1, 0, 0] ]
kash> u := L[1]; v:= L[2];
[1, 0, 1, 0]
[1, -1, 0, 0]
kash> EltUnitDecompose(u*v);
[ [ [1, 0, 1, 0], 1 ], [ [1, -1, 0, 0], 1 ] ]
kash> EltUnitDecompose(u*v,"expons");
[ 0, 1, 1 ]
kash> EltUnitDecompose(-u*v);
[ [ -1, 1 ], [ [1, 0, 1, 0], 1 ], [ [1, -1, 0, 0], 1 ] ]
kash> EltUnitDecompose(-u*v,"expons");
[ 1, 1, 1 ]
kash> EltUnitDecompose(u^3*v^2);
[ [ [1, 0, 1, 0], 3 ], [ [1, -1, 0, 0], 2 ] ]
kash> EltUnitDecompose(u^3*v^2,"expons");
> [ 0, 3, 2 ]

</pre>

<p>


<hr>

<p>
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