Difference between revisions of "ApCoCoA-1:CharP.GBasisF8"

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<command>
 
     <title>CharP.GBasisF8</title>
 
     <title>CharP.GBasisF8</title>
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     <seealso>
 
     <seealso>
       <see>GBasis</see>
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       <see>ApCoCoA-1:GBasis|GBasis</see>
     <see>Introduction to CoCoAServer</see>
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     <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
     <see>Introduction to Groebner Basis in CoCoA</see>
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     <see>ApCoCoA-1:Introduction to Groebner Basis in CoCoA|Introduction to Groebner Basis in CoCoA</see>
     <see>CharP.GBasisF2</see>
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     <see>ApCoCoA-1:CharP.GBasisF2|CharP.GBasisF2</see>
     <see>CharP.GBasisF4</see>
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     <see>ApCoCoA-1:CharP.GBasisF4|CharP.GBasisF4</see>
     <see>CharP.GBasisF16</see>
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     <see>ApCoCoA-1:CharP.GBasisF16|CharP.GBasisF16</see>
     <see>CharP.GBasisF32</see>
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     <see>ApCoCoA-1:CharP.GBasisF32|CharP.GBasisF32</see>
     <see>CharP.GBasisF64</see>
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     <see>ApCoCoA-1:CharP.GBasisF64|CharP.GBasisF64</see>
     <see>CharP.GBasisF128</see>
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     <see>ApCoCoA-1:CharP.GBasisF128|CharP.GBasisF128</see>
     <see>CharP.GBasisF256</see>
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     <see>ApCoCoA-1:CharP.GBasisF256|CharP.GBasisF256</see>
     <see>CharP.GBasisF512</see>
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     <see>ApCoCoA-1:CharP.GBasisF512|CharP.GBasisF512</see>
     <see>CharP.GBasisF1024</see>
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     <see>ApCoCoA-1:CharP.GBasisF1024|CharP.GBasisF1024</see>
     <see>CharP.GBasisF2048</see>
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     <see>ApCoCoA-1:CharP.GBasisF2048|CharP.GBasisF2048</see>
     <see>CharP.GBasisModSquares</see>
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     <see>ApCoCoA-1:CharP.GBasisModSquares|CharP.GBasisModSquares</see>
<see>Representation of finite fields</see>
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<see>ApCoCoA-1:Representation of finite fields|Representation of finite fields</see>
 
   </seealso>
 
   </seealso>
  
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     <key>charP.GBasisF8</key>
 
     <key>charP.GBasisF8</key>
 
     <key>finite field</key>
 
     <key>finite field</key>
     <wiki-category>Package_charP</wiki-category>
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     <wiki-category>ApCoCoA-1:Package_charP</wiki-category>
 
   </command>
 
   </command>

Latest revision as of 09:55, 7 October 2020

This article is about a function from ApCoCoA-1.

CharP.GBasisF8

Computing a Groebner Basis of a given ideal in F_8.

Syntax

CharP.GBasisF8(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 F_8 = (Z/(2))[x]/(x^3 + x +1).

  • @param Ideal An Ideal in a Ring over Z, where the elements 0,...,7 represent the elements of the field F_8. 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.

Example

Use R::=QQ[x,y,z];
I:=Ideal(x-y^2,x^2+xy,y^3);
GBasis(I);

[x^2 + xy, -y^2 + x, -xy]
-------------------------------
Use Z::=ZZ[x,y,z];
-- WARNING: Coeffs are not in a field
-- GBasis-related computations could fail to terminate or be wrong

-------------------------------
I:=Ideal(x-y^2,x^2+xy,y^3);
CharP.GBasisF8(I);
-- WARNING: Coeffs are not in a field
-- GBasis-related computations could fail to terminate or be wrong
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
[y^2 + 4x, x^2, xy]
-------------------------------


See also

GBasis

Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

CharP.GBasisF2

CharP.GBasisF4

CharP.GBasisF16

CharP.GBasisF32

CharP.GBasisF64

CharP.GBasisF128

CharP.GBasisF256

CharP.GBasisF512

CharP.GBasisF1024

CharP.GBasisF2048

CharP.GBasisModSquares

Representation of finite fields