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<h2><a name="OrderUnitsExcep"></a>
OrderUnitsExcep
</h2>

Computes all exceptional units of an order.

<p><p>

<h3>
Syntax:
</h3>
<pre>L := OrderUnitsExcep(o);
L := OrderUnitsExcep(o,"orbits"|"list");
n := OrderUnitsExcep(o,"number");</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  L  </pre></td>
<td></td>
</tr>
<tr>
<td>integer</td>
<td><pre>  n  </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#EltExcepUnitOrbit">EltExcepUnitOrbit</a>, <a href="kashref.html#OrderExcepSequence">OrderExcepSequence</a>, <a href="kashref.html#OrderUnitsEquation">OrderUnitsEquation</a>

<p>

<h3>
Description:
</h3>
Let o be an arbitrary order. In the terminology of T.~Nagell
Nag a unit \varepsilon \in o is exceptional if 1
- \varepsilon is also a unit. \smallskip

For an exceptional unit \varepsilon \in o all the units
 \varepsilon_1 = \varepsilon, quad
\varepsilon_2 = frac{1}{\varepsilon}, quad
\varepsilon_3 = 1-\varepsilon, quad
\varepsilon_4 = frac{1}{1-\varepsilon}, quad
\varepsilon_5 = frac{\varepsilon-1}{\varepsilon}, quad
\varepsilon_6 = frac{\varepsilon}{\varepsilon-1}

are exceptional. The set formed by
\varepsilon_1, &hellip; ,\varepsilon_6 is called the orbit
of \varepsilon. \bigskip

<tt> OrderUnitsExcep(o)</tt> or <tt> OrderUnitsExcep(o,"orbits")</tt>
returns a list whose entries represent all orbits.\medskip

<tt> OrderUnitsExcep(o,"number")</tt>
returns the number of exceptional units in o.\medskip

<tt> OrderUnitsExcep(o,"list")</tt>
returns the list <tt> L</tt> of all exceptional units in o.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Compute exceptional units in Z[zeta_7]. 
<pre>

kash> O := Order(Poly(Zx,[1,1,1,1,1,1,1]));
Generating polynomial: x^6 + x^5 + x^4 + x^3 + x^2 + x + 1

kash> OrderUnitsExcep(O);
[ [0, -1, 0, 0, 0, 0], [0, -1, -1, 0, 0, 0], [0, 0, -1, 0, 0, 0], 
  [0, -1, -1, -1, 0, 0], [0, -1, -1, -1, -1, 0], [-1, 0, 0, -1, -1, 0], 
  [0, 0, 0, -1, -1, 0], [1, 0, 0, -1, -1, 0], 
  [-684, 0, -549, -244, -244, -549], [-7, 0, -6, -3, -3, -6], 
  [-3, 0, -3, -1, -1, -3], [-2, 0, -2, -1, -1, -2] ]
kash> OrderUnitsExcep(O,"number");
72
kash> OrderUnitsExcep(O,"list");
> [ [0, -1, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0], [0, -1, -1, -1, -1, -1], 
  [0, -1, 0, -1, 0, -1], [1, 1, 0, 1, 0, 1], [1, 1, 1, 1, 1, 1], 
  [0, -1, -1, 0, 0, 0], [1, 1, 1, 0, 0, 0], [0, 0, -1, 0, -1, 0], 
  [0, -1, 0, 0, -1, 0], [1, 1, 0, 0, 1, 0], [1, 0, 1, 0, 1, 0], 
  [0, 0, -1, 0, 0, 0], [1, 0, 1, 0, 0, 0], [0, -1, 0, 0, -1, -1], 
  [0, 0, 0, 0, 0, -1], [1, 0, 0, 0, 0, 1], [1, 1, 0, 0, 1, 1], 
  [0, -1, -1, -1, 0, 0], [0, 0, 0, -1, 0, 0], [1, 0, 0, 1, 0, 0], 
  [1, 1, 1, 1, 0, 0], [0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 1, 0], 
  [0, -1, -1, -1, -1, 0], [1, 1, 1, 1, 1, 0], [0, -1, -1, 0, -1, -1], 
  [0, 0, 0, -1, 0, -1], [1, 0, 0, 1, 0, 1], [1, 1, 1, 0, 1, 1], 
  [-1, 0, 0, -1, -1, 0], [2, 0, 0, 1, 1, 0], [1, 0, -1, -2, -2, -1], 
  [-1, 0, -1, -1, -1, -1], [2, 0, 1, 1, 1, 1], [0, 0, 1, 2, 2, 1], 
  [0, 0, 0, -1, -1, 0], [1, 0, 0, 1, 1, 0], [0, 0, -1, -1, -1, -1], 
  [0, 0, -1, 0, 0, -1], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 1], 
  [1, 0, 0, -1, -1, 0], [0, 0, 0, 1, 1, 0], [-1, 0, -1, 0, 0, -1], 
  [2, 0, -1, 1, 1, -1], [-1, 0, 1, -1, -1, 1], [2, 0, 1, 0, 0, 1], 
  [-684, 0, -549, -244, -244, -549], [136, 0, -305, -549, -549, -305], 
  [441, 0, -244, 305, 305, -244], [-440, 0, 244, -305, -305, 244], 
  [-135, 0, 305, 549, 549, 305], [685, 0, 549, 244, 244, 549], 
  [-7, 0, -6, -3, -3, -6], [2, 0, -3, -6, -6, -3], [5, 0, -3, 3, 3, -3], 
  [-4, 0, 3, -3, -3, 3], [-1, 0, 3, 6, 6, 3], [8, 0, 6, 3, 3, 6], 
  [-3, 0, -3, -1, -1, -3], [1, 0, -2, -3, -3, -2], [3, 0, -1, 2, 2, -1], 
  [-2, 0, 1, -2, -2, 1], [0, 0, 2, 3, 3, 2], [4, 0, 3, 1, 1, 3], 
  [-2, 0, -2, -1, -1, -2], [1, 0, -1, -1, -1, -1], [1, 0, -1, 0, 0, -1], 
  [0, 0, 1, 0, 0, 1], [0, 0, 1, 1, 1, 1], [3, 0, 2, 1, 1, 2] ]

</pre>

<p>


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