[GAP Forum] arithmetic in C

[Wolfgang Lindner]

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 of the embedded 'gapplets').

best
Wolfgang


-----Ursprüngliche Nachricht-----
Von: Max Horn <max at quendi.de>
Cc: GAP Forum <forum at 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
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