Subextension over non-prime subfield as a quotient

savecancel

Hi!

I am looking for a light and easy way to build the following objects:

• a finite (possibly non prime) field Fq;
• an irreducible polynomial p in Fq[X];
• and a finite extension L of Fq which contains L0 = Fq[X]/p as a subfield.

Crucially, I want to be able to manipulate the generator z = L0(X) (i.e. the image of X in L0) as an element of L. I also need to view these extensions as extension over Fq (e.g. using .over(Fq)) and not the prime subfield (in particular, I do not want to see L as an extension of L0).

Let me know if this is not clear and you need more explanations.

For this, I can define my extensions as follows:

sage: Fq = GF(7^2)
sage: FqX.<X> = Fq[]
sage: p = FqX.irreducible_element(3)
sage: L0 = Fq.extension(modulus=p).over(Fq); L0
Univariate Quotient Polynomial Ring in X over Finite Field in z2 of size 7^2 with modulus X^3 + z2*X + z2 + 3 over its base
sage: L = Fq.extension(modulus=FqX.irreducible_element(6)).over(Fq); L
Univariate Quotient Polynomial Ring in X over Finite Field in z2 of size 7^2 with modulus X^6 + (6*z2 + 4)*X + 4*z2 + 6 over its base

Now I can define z = L0(X):

sage: z = L0(X)
sage: assert z == L0.gen()
sage:

But I can't manage to see z as an element of L. In the following snippet, it seems that casting z as an element of L converts it to a generator of L:

sage: zL = L(z)
sage: zL == L.gen()
True

I tried other ways to create the extensions. Sometimes coercion is a problem. Sometimes some methods are available on a method but not the other. Does anybody has any idea?