Re: polgalois

Cher Bill,

génial, ca marche et suffit tt a fait!

Considère-toi invité pour un pot (voir plus) lors de ta prochaine
visite-conférence à Heidelberg, eg. pour le HLF Laureate Forum ;-)

merci mille fois!

Harald

+49 151 611 711 33

Am Do., 26. Okt. 2023 um 14:18 Uhr schrieb Bill Allombert <
[email protected]>:

> 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
>