Hello all,
Don't mean to reply to my own question. I guess what I am wondering is
why the finite extension of a finite field isn't a finite field. By
having the results being a polynomial ring over the prime field it
seems as if a lot is lost. Is there room to alter this behavior, or
was there a conscious design decision made when choosing this? Once
again thank you for all of your hard work.
David
On May 25, 12:26 pm, "D. Monarres" <dmmonar...@gmail.com> wrote:
> Hello All,
>
> First off, I would like to say that Sage is great and all of the hard
> work that the developers have put forth definitely shows.
>
> I am writing a tutorial for using Sage in undergraduate mathematics
> courses at a public 4 year university and am running into a few
> issues. I would appreciate any help and if these are indeed bugs than
> I wouldn't mind being pointed toward where I can go to help fix them.
>
> So right now I am going through constructing finite fields in a couple
> of different ways. One using the build in GF command with different
> moduli and the other is by constructing a prime field and then
> extending it using the root of a irreducible polynomial of specified
> degree. As there is not any "all irreducible polynomials of a certain
> degree command in Sage" I just constructed all polynomials of that
> degree and then filtered the list using the is_reducible() method.
>
> For example, to construct all degree two polynomials over GF(5) I did
>
> sage: F5 = GF(5)
> sage: P.<x> = PolynomialRing(F5, 'x')
> sage: AP = [ a1 + a2*x + a3*x^2 for (a1,a2,a3) in F5^3 if a3 !=
> F5(0) ]
>
> then I filter like this
>
> sage: IR = [ p for p in AP if p.is_irreducible() ]
> sage: PR = [ p for p in AP if p.is_primitive() ]
>
> Then I construct F_{5^2}
>
> sage: F25.<a> = F5.extension( PR[0], 'a')
>
> and everything works great. The problem arises when I want to extend
> this field. When I try to construct my polynomial over F25 I get an
> error that F25 does not allow for iteration.
>
> Is there a better way to construct a list of irreducible and/or
> primitive polynomials in Sage? Do you think that
> PolynomialQuotientRing_field can be extended to support iteration as
> long as it is finite? I would be willing to make an attempt at doing
> this if somebody could give me some tips as to where to begin?
>
> Thank you all for your hard work,
>
> DavidMonarres
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