[GAP Forum] Bijection from k^n to the extension of k of degree n

[Huta Gana]

Dear GAP forum,

I would like to ask some questions relating to the construction of the
bijection from k^n to  the extension of k of degree n. I have used the
following codes



*gap> k:=GF(2^2);;gap> x:=X(k,"x");; g:=x^3+x+1;;       gap>
Kg:=AlgebraicExtension(k,g);;*

Then, an element in Kg has a form: a+bt+bt^2  (a, b, c in *k*, g(t)=0). I
want to construct the bijection T: *k*^3->Kg is define by
(a,b,c)->a+bt+bt^2.

As we know that *k*^3 and Kg are 3-dimesional vector space over over *k*.
To consider Kg as a *k*-vector space, I use


*gap> Kg:=AsVectorSpace(k,Kg);<algebra-with-one of dimension 6 over
GF(2^2)>*

 So, we can build an isomorphism from *k*^3 to K. But


*gap> Size(Basis(Kg));*



*6                                    //not 3gap> Size(Basis(k^3));3*
>From that, I can not construct an isomorphism from to *k*^3 ->Kg.



*gap>f:=LeftModuleGeneralMappingByImages(k^3,Kg,Basis(k^3),Basis(Kg));Error,
<gens> and <imgs> must have the same length*

Could you help me how to construct the the bijection T: *k*^3->Kg?

Thank you very much.

Best regards,

Huta