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Basis of extension

asked 2015-06-07 15:34:22 -0500

adhalanay gravatar image

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?

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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.

B r u n o gravatar imageB r u n o ( 2015-06-08 04:43:15 -0500 )edit

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

adhalanay gravatar imageadhalanay ( 2015-06-08 13:30:15 -0500 )edit

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answered 2015-06-09 12:43:23 -0500

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

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Asked: 2015-06-07 15:34:22 -0500

Seen: 269 times

Last updated: Jun 09 '15