Luiz Figueiredo asked:
I am new to GAP and haven't really gone over all the documentation
yet. I would like to know if GAP can be used to determine the
ramification groups of a prime in a Galois extension of the rationals
defined as the splitting field of a given polynomial.
Not really. GAP only supports some basic commands to compute or identify
the Galois group of algebraic extensions and to compute subfields.
(see the manual sections `GaloisGroup', `GaloisType' and 'DecomPoly')
but nothing as involved as ramification groups. I'm also not aware of
any user who has written own routines for this purpose.
As our resources are limited, we are not planning to extend the functions
for algebraic number theory in the forseeable future. However there are other
systems designed especially for number theory which might provide such
routines: the Pari/GP system developed by the group of H.Cohen in Bordeaux
(anonymous ftp to megrez.ceremab.u-bordeaux.fr, for information write to
firstname.lastname@example.org); the KANT system developed by the group of
M.Pohst in Berlin (ftp.math.tu-berlin.de/pub/algebra/Kant, for information
email@example.com) whose shell is very similar to GAP
and the Simath system developed by the group of H.Zimmer in Saarbr"ucken
(ftp.math.uni-sb.de, for information firstname.lastname@example.org).
Finally one remark aside: If any reader is aware of a reasonable algorithm to
compute ramification groups (Cohen's book does not mention this problem) I'd
be interested to hear about it by personal mail. It seems to me that
computing these groups is at least as complicated as to computing the Galois
group as a group of field automorphisms which is already quite hard.
Alexander Hulpke (email@example.com)