Re: polgalois

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.. ;-)

Anyway, I tried out your commands with a simple example for n=12,

but some index issue cropped up, Any idea what causes it?

? G=galoissplittinginit(x^12+3); \\ not S5
S=galoissubgroups(G);
H=select(s->vecprod(Vec(s[2]))==20,S)[1];
P=galoisfixedfield(G,H,1); 
polgalois(P)

*** nonexistent component: index > 0

Cheers,
Harald

Am Do., 26. Okt. 2023 um 10:17 Uhr schrieb Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>:

On Thu, Oct 26, 2023 at 07:57:13AM +0200, Harald Borner wrote:
> Bonjour,
>
> Félicitations a tous pour Pari/GP - un super package, easy to use!
>
> Une Q: avez-vous des plans pour une extension de la fonction
> polgalois
> aux degrées n>11, eg. 12-15 ou 16?? Ca serait génial...

Yes, we have been working on this for a long time, but this still does not
quite work as we hoped. We have trouble with the group-theoretical part
of the algorithm.

Also we plan for the new polgalois to return the action of the Galois group
on the roots so that one could compute the corresponding fixed field.

For groups of small order (says <=1000), one option is galoissplittinginit
which computes the Galois group of the splitting field of the polynomial.

G=galoissplittinginit(x^5-x-1); \\ S5
S=galoissubgroups(G);
H=select(s->vecprod(Vec(s[2]))==20,S)[1];
P=galoisfixedfield(G,H,1)
\\%4 = x^6+3600*x^5+8172480*x^4+14907176960*x^3+16934023065600*x^2+71434160884383744*x+86015979642640138240
polgalois(P)
\\%5 = [120,-1,1,"L(6):2 = PGL(2,5) = S_5(6)"]

Cheers,
Bill