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Basis of extension

asked 2015-06-07 22:34:22 +0200

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


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


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 11:43:15 +0200 )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 20:30:15 +0200 )edit

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answered 2015-06-09 19:43:23 +0200

vdelecroix gravatar image

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).


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Asked: 2015-06-07 22:34:22 +0200

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Last updated: Jun 09 '15