> < ^ Date: Mon, 14 Aug 1995 13:35:00 +1553
> < ^ From: Alexander Hulpke <hulpke@math.colostate.edu >
> < ^ Subject: Re: Galois function

Dear Forum,

Luiz Figueiredo asked:

I have been trying to use the function Galois(p), for p polynomial to
get the Galois group of the splitting field of p. This function
returns an integer, the Galois type of the Galois group.
Can anyone tell me where there is a list of all Galois types for a
given degree, ie. which group correspond to which Galois types?

This is the index in the list of transitive groups. Thus if 'Galois' returns
'n' for an irreducible polynomial of degree 'd' then
'TransitiveGroup(d,n)' returns a permutation group which is permutation
isomorphic to the operation of the Galois group on the roots.
(The function works only for irreducible polynomials, accordingly the Galois
group acts transitively).
The arrangement is the same as in the article

         author  = "Gregory Butler and John McKay",
         title   = "The transitive groups of degree up to 11",
         journal = CommAlg,
         volume  = 11,
         year    = 1983,
         pages   = "863--911"

There is no facility to deduce from this the actual action on the roots
when embedding the splitting field say in the complex numbers.

The groups obtained via 'TransitiveGroup' already have names in GAP (i.e.
TransitiveGroup(6,10) has the name "1/2[S(3)^2]2". These names however
are not yet 'cleaned up' and will change. They should not be used to refer
to these groups in the literature. (I have a newer list with
better names I can provide you with if you want to use names.)

(Your question seems to indicate that you still lack the manual section on
'GaloisType' and accordingly did not do the upgrade to version 3.4.2. As these
patches also fix at least one bug in the 'Galois' code you should seriously
consider upgrading.)

> Thanks,
You're welcome,

Alexander Hulpke

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