Re: polgalois

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Thu, Oct 26, 2023 at 11:40:59AM +0200, Harald Borner wrote:
> Cher Bill,
> 
> merci bcp. pour ta réponse rapido!!
> 
> ->  For groups of small order (says <=1000), one option is
> galoissplittinginit
> which computes the Galois group of the splitting field of the polynomial.
> 
> This would be fantastic! more than enough for what we need.
> When you say "the" splitting field, I guess you mean not just any such, but
> the smallest possible.
> What is the difference then to what polgalois returns, if not the Gal grp.
> of a/the smallest splitting field?
> (just came back from Nepal, so I have to get my mind back into the nitty
> gritty of Galois theory.. ;-)

For your example, you can do this:

? G=galoissplittinginit(x^12+3);
? #G.group
%2 = 24
? galoisidentify(G)
%3 = [24,8]
? galoisexport(G)
%4 = "Group((1, 11, 10, 24, 14, 15)(2, 22, 21, 23, 3, 4)(5, 17, 12, 20, 8, 13)(6, 16, 7, 19, 9, 18), (1, 6, 24, 19)(2, 17, 23, 8)(3, 20, 22, 5)(4, 12, 21, 13)(7, 11, 18, 14)(9, 15, 16, 10), (1, 2, 10, 21, 14, 3)(4, 11, 22, 24, 23, 15)(5, 7, 12, 9, 8, 6)(13, 16, 17, 19, 20, 18))"

%2 tells you the group has 24 elements.
%3 tells you the group is isomorphic as an abstract group to SmallGroup(24,8)
%4 gives you the associated permutation group in GAP syntax.

Note: PARI galoisidentify only works for groups of order <=127, because going further require huge tables.

Then in GAP you can do

gap> G:=Group((1, 11, 10, 24, 14, 15)(2, 22, 21, 23, 3, 4)(5, 17, 12, 20, 8, 13)(6, 16, 7, 19, 9, 18), (1, 6, 24, 19)(2, 17, 23, 8)(3, 20, 22, 5)(4, 12, 21, 13)(7, 11, 18, 14)(9, 15, 16, 10), (1, 2, 10, 21, 14, 3)(4, 11, 22, 24, 23, 15)(5, 7, 12, 9, 8, 6)(13, 16, 17, 19, 20, 18));
gap> TransitiveIdentification(G);
14
gap> IdGroup(G);
[ 24, 8 ]
gap> TransitiveGroup(12,14);
D(4)[x]C(3)

So your group is D(4)[x]C(3)

Cheers,
Bill