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,
David Monarres
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