Difference between revisions of "ApCoCoA-1:CharP.GBasisF4"

From ApCoCoAWiki
(Added ApCoCoAServer note)
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<command>
 
<command>
 
     <title>Char2.GBasisF4</title>
 
     <title>Char2.GBasisF4</title>
     <short_description>computing a gbasis of a given ideal in <formula>\mathbb{F}_{4}</formula></short_description>
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     <short_description>Computing a Groebner Basis of a given ideal in <formula>\mathbb{F}_{4}</formula>.</short_description>
 
<syntax>
 
<syntax>
$char2.GBasisF4(Ideal):List
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Char2.GBasisF4(Ideal:IDEAL):LIST
 
</syntax>
 
</syntax>
 
     <description>
 
     <description>
{{ApCoCoAServer}}
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<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
 +
<par/>
 +
This command computes a Groebner basis in the field <formula> \mathbb{F}_{4} = (\mathbb{Z}_{\setminus(2)} [x])_{\setminus(x^2 + x +1)}</formula>.
  
This command computes a Groebner basis in the field <formula> \mathbb{F}_{4} = (\mathbb{Z}_{\setminus(2)} [x])_{\setminus(x^2 + x +1)}</formula>. It uses the ApCoCoA Server and the ApCoCoALib's class [[ApCoCoALib:RingF4|RingF4]].
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<itemize>
 
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<item>@param <em>Ideal</em> An Ideal in a Ring over Z, where the elements 0,...,3 represent the field's elements. For short, the binary representation of the number represents the coefficient vector if the polynomial in the field, e.g. 11 = 8 + 2 + 1 = 2^3 + 2^1 + 2^0. So the number 11 corresponds to the polynomial x^3 + x + 1.</item>
The command's input is a an Ideal in a Ring over Z, where the elements 0,..., 3 represent the field's elements. Details on this representation can be found [[ApCoCoA:Representation_of_finite_fields|here]]. For short, the binary representation of the number represents the coefficient vector if the polynomial in the field, e.g.  
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<item>@return A Groebner Basis of the given ideal.</item>
<formula> 11 = 8 + 2 + 1 = 2^3 + 2^1 + 2^0</formula>
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</itemize>
So the number <formula>11</formula> corresponds to the polynomial <formula>x^3 + x + 1</formula>.
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    </description>
  
    </description>
 
 
     <seealso>
 
     <seealso>
 
       <see>GBasis</see>
 
       <see>GBasis</see>
     <see>char2.GBasisF2</see>
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     <see>Char2.GBasisF2</see>
     <see>char2.GBasisF8</see>
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     <see>Char2.GBasisF8</see>
     <see>char2.GBasisF16</see>
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     <see>Char2.GBasisF16</see>
     <see>char2.GBasisF32</see>
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     <see>Char2.GBasisF32</see>
     <see>char2.GBasisF64</see>
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     <see>Char2.GBasisF64</see>
     <see>char2.GBasisF128</see>
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     <see>Char2.GBasisF128</see>
     <see>char2.GBasisF256</see>
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     <see>Char2.GBasisF256</see>
     <see>char2.GBasisF512</see>
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     <see>Char2.GBasisF512</see>
     <see>char2.GBasisF1024</see>
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     <see>Char2.GBasisF1024</see>
     <see>char2.GBasisF2048</see>
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     <see>Char2.GBasisF2048</see>
     <see>char2.GBasisF4096</see>
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     <see>Char2.GBasisF4096</see>
     <see>char2.GBasisModSquares</see>
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     <see>Char2.GBasisModSquares</see>
 
 
 
   </seealso>
 
   </seealso>
     <key>heldt</key>
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 +
     <key>GBasisF4</key>
 
     <key>char2.GBasisF4</key>
 
     <key>char2.GBasisF4</key>
 
     <wiki-category>Package_char2</wiki-category>
 
     <wiki-category>Package_char2</wiki-category>
 
   </command>
 
   </command>

Revision as of 07:00, 23 April 2009

Char2.GBasisF4

Computing a Groebner Basis of a given ideal in <formula>\mathbb{F}_{4}</formula>.

Syntax

Char2.GBasisF4(Ideal:IDEAL):LIST

Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

This command computes a Groebner basis in the field <formula> \mathbb{F}_{4} = (\mathbb{Z}_{\setminus(2)} [x])_{\setminus(x^2 + x +1)}</formula>.

  • @param Ideal An Ideal in a Ring over Z, where the elements 0,...,3 represent the field's elements. For short, the binary representation of the number represents the coefficient vector if the polynomial in the field, e.g. 11 = 8 + 2 + 1 = 2^3 + 2^1 + 2^0. So the number 11 corresponds to the polynomial x^3 + x + 1.

  • @return A Groebner Basis of the given ideal.

See also

GBasis

Char2.GBasisF2

Char2.GBasisF8

Char2.GBasisF16

Char2.GBasisF32

Char2.GBasisF64

Char2.GBasisF128

Char2.GBasisF256

Char2.GBasisF512

Char2.GBasisF1024

Char2.GBasisF2048

Char2.GBasisF4096

Char2.GBasisModSquares