# ApCoCoA-1:CharP.GBasisF4

(Redirected from ApCoCoA-1:Charp.GBasisF4)

## CharP.GBasisF4

Computing a Groebner Basis of a given ideal in F_4.

### Syntax

```CharP.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 F_4 = (Z/(2))[x]/(x^2 + x +1).

• @param Ideal An Ideal in a Ring over Z, where the elements 0,...,3 represent the elements of the field F_4. 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.GBasisF4(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 + 2x, x^2, xy]
-------------------------------
```