# Basis of extension

Assume that $q$ is a power of a prime number. Consider the field extension $F_q \subset F_{q^n}$ . Both fields can be thought as subfields of the algebraic closure of $F_p$, defined with

K=GF(p).algebraic_closure()


The question is: Is it possible to compute a basis of $F_{q^n}$ over $F_{q}$ ? Or at least to have something in the spirit of

K.<x1,...,xn>=GF(q)[]


which can be done when $q$ is prime?

edit retag close merge delete

What do you mean by a basis? Suppose you have sage: K = GF(q,'a'). Then you can type sage: (L,f) = K.extension(2,'b',map=True) to get in L the finite field with q^2 elements, defined as an extension K. And f gives you the embedding of K inside L.

( 2015-06-08 04:43:15 -0600 )edit

I am interested in finding $n$ elements of $F_{q^n}$ linear-independent over $F_q$, even when $q$ is not prime.

( 2015-06-08 13:30:15 -0600 )edit

Sort by » oldest newest most voted

You can always do find these elements

sage: a = K.multiplicative_generator()
sage: basis = [a**i for i in range(dim)]


where K is your finite field and dim is the dimension of your vector space (or the codimension of the ground field sitting inside the big one).

Vincent

more