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

Computation of Galois groups.

<p><p>

<h3>
Syntax:
</h3>
<pre>Galois ( f [,p] [, "fast"] );
Galois ( o [,p] [, "fast"] );

Galois ( f , "complex" [, n] );
Galois ( o , "complex" [, n] );

GaloisT ( f [,p] [, "fast"] );
GaloisT ( o [,p] [, "fast"] );

GaloisT ( f , "complex" [, n] );
GaloisT ( o , "complex" [, n] );</pre>

<p>

<table>
<tr>
<td>polynomial</td>
<td><pre>  f  </pre></td>
<td></td>
</tr>
<tr>
<td>order</td>
<td><pre>  o  </pre></td>
<td></td>
</tr>
<tr>
<td>prime number</td>
<td><pre>  p  </pre></td>
<td></td>
</tr>
<tr>
<td>positive integer</td>
<td><pre>  n  </pre></td>
<td>precision</td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#GaloisGlobals">GaloisGlobals</a>, <a href="kashref.html#GaloisGroupsPossible">GaloisGroupsPossible</a>, <a href="kashref.html#GaloisModulo">GaloisModulo</a>, <a href="kashref.html#GaloisTree">GaloisTree</a>, <a href="kashref.html#GaloisRoots">GaloisRoots</a>, <a href="kashref.html#GaloisNumberToName">GaloisNumberToName</a>, <a href="kashref.html#GaloisBlocks">GaloisBlocks</a>

<p>

<h3>
Description:
</h3>
This function computes the Galois group of an irreducible
polynomial with coefficients in Q and with degree up to 23.

The first two callings are a p-adic version of the method of
Stauduhar.
The optional integer p gives the prime number to be used for
the p-adic computations. Not using the parameter <tt> "fast"</tt>
the p-adic version returns a proven result.
The optional parameter <tt> "fast"</tt> is able
to speed up the computation considerably, but incorrect results
may occure. The option <tt> "fast"</tt> makes use of lower bounds
during the inclusion test of a Galois group computation.
The next two callings are the complex version of the method of
Stauduhar. The optional integer n gives the precision to be used.
The minimal precision is 100. \medskip

<tt> Galois()</tt> returns the name of the Galois group
whereas
<tt> GaloisT()</tt> returns its number in T-notation as
classified in \medskip GAP GAP.

The implementation of the <tt> Galois</tt> function Ge1 in KASH\
bases on the algorithm of R.P.Stauduhar
Stau.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>

<pre>

kash> Galois(x^12-2);
"D(4)[x]S(3)"
kash> o := Order(Z, 12, 2);
Generating polynomial: x^12 - 2

kash> GaloisModulo(o, 100);
[ 28, 81, 83, 86, 125, 134, 141, 143, 156, 185, 186, 193, 208, 209, 213, 217, 
  222, 239, 240, 248, 250, 258, 260, 267, 268, 270, 274, 281, 283, 288, 289, 
  292, 293, 294, 299, 301 ]
kash> f := x^15 + 2*x^9 - 5*x^6 + 2*x^3 - 1;
x^15 + 2*x^9 - 5*x^6 + 2*x^3 - 1
kash> GaloisT(f);
> 61

</pre>

<p>


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