dear Max, no, there was no private answer yet .. so I appeciate your hints very much. There are informative, concrete and helpful.
I'm reading the nice book of Cohen et al. 'Algebra interactive' and try to follow their constructions making _direct_ use of GAP (and not the embedded gapplets). best Wolfgang -----Ursprüngliche Nachricht----- Von: Max Horn <m...@quendi.de> Cc: GAP Forum <fo...@gap-system.org> Datum: Dienstag, 20. März 2012 11:14 Betreff: Re: [GAP Forum] arithmetic in C |Dear Wolfgang, | |I am not sure if you perhaps already got a private reply to your email. If not, maybe the following will help you. | |Am 14.03.2012 um 09:05 schrieb Wolfgang Lindner: | |> dear group, |> |> I know how to calculate with Rationals, ZmodnZ, Integers etc. |> But I could not find infos in the help-index of GAP |> how to do calculations in the complex field C |> (I know about gaussionInteger). | | |Short answer: Use "Cyclotomics" or one of its subfields. Make sure to read the GAP manual on them and on abelian number fields: | <http://www.gap-system.org/Manuals/doc/htm/ref/CHAP018.htm> | <http://www.gap-system.org/Manuals/doc/htm/ref/CHAP058.htm> | |The following might also be of interest: | <http://www.gap-system.org/Manuals/doc/htm/ref/CHAP056.htm> | <http://www.gap-system.org/Manuals/doc/htm/ref/CHAP065.htm> | | |Long answer: It is essentially impossible to compute with the "full set" of complex numbers (or real numbers) on a computer; in particular, not every real (and hence not every complex) number is computable (see e.g. <http://en.wikipedia.org/wiki/Computable_number>). | |But for the vast majority of cases (at least in my personal experience), one doesn't really need the full set of real or complex numbers; rather, one only needs to deal with a few select numbers, such as "square root of 2" or "pi". GAP allows you to work with the former: One can construct abelian extension fields of the rational numbers in GAP, which all are subfields of the "field of cyclotomic numbers". So the following works: | |gap> Sqrt(-1); |E(4) |gap> Sqrt(2); |E(8)-E(8)^3 | |However, this does not allow you to work with pi directly, as that lives in a transcendental extension. There are some tricks to deal with that to a certain extent. Note that pi and similar transcendentals seem not to be really necessary to do group theory, which is probably why they are not supported as such. | | |> I would like to work in C[X] etc. | |gap> R:=PolynomialRing(Cyclotomics, "x"); |Cyclotomics[x] |gap> x:=R.1; |x |gap> f:=x^2-x+1; |x^2-x+1 | | |Cheers, |Max |_______________________________________________ |Forum mailing list |Forum@mail.gap-system.org |http://mail.gap-system.org/mailman/listinfo/forum _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
