On Jan 11, 7:19 am, javier <vengor...@gmail.com> wrote:
[...]
> I have been trying some code to convert elements of gap finite fields
> into the corresponding elements of sage finite fields. The sort of
> straightforward manner fails because of this behavior:
>
> sage: a = gap("Z(2^4)")
> sage: a^5
> Z(2^2)
>
> and apparently sage has no way of coercing finite fields into bigger
> finite fields (in this example a coercion map from GF(4) to GF(16)
> would send z4 to z16^5):
[...]
> TypeError: unable to coerce from a finite field other than the prime
> subfield
I reckon this must be due to Sage representing the finite field of
order p^n
as quotient rings F_p[x]/(f(x)), with f an irreducible polynomial of
degree n. Indeed, in this case to do the coercion to, say F_{p^m}=F_p
[x]/(g(x)), (with n dividing m, of course) would require some
nontrivial (and slow) polynomial arithmetic, unless f and g are
somehow "related".GAP does it in a more clever way, at least for its "standard" finite fields; f and g are always related in such a way that such a computation is trivial. Namely, the primitive element Z(p^n) of the 1st field equals Z(p^m)^((p^m-1)/(p^n-1)), where Z(p^m) is the primitive element of the 2nd field. > > I can find a way around this by parsing the GAP strings with a lot of > care, but that is going to be an ugly piece of code and I'd rather > avoid it. Is there an easy way of getting the coercion maps for finite > fields, or maybe a way of regularizing the GAP output so that > everything is written in terms of the generator? for the latter, hopefully the remark above is useful. Dmitrii > > Cheers > J
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