Re: peculiar galoisinit() behavior

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

On Fri, Jun 11, 2004 at 07:13:35PM -0400, Igor Schein wrote:
> Hi,
> 
> using latest CVS,
> 
> {galoisinit(
> x^64 - 224*x^62 + 23552*x^60 - 1546848*x^58 + 71168328*x^56 - 2436833952*x^5
> 4 + 64407288512*x^52 - 1346248726816*x^50 + 22638605928092*x^48 - 3100404236
> 90592*x^46 + 3482991142252160*x^44 - 32107751441411296*x^42 + 24044849215906
> 9432*x^40 - 1419156581985832480*x^38 + 6021441872671174464*x^36 - 1123013514
> 6026525344*x^34 - 82256825364060916154*x^32 + 1006806032298827456352*x^30 - 
> 5675649698314152988928*x^28 + 16929909843292332382944*x^26 + 514728199833124
> 8762616*x^24 - 319192745820875674502624*x^22 + 1645617652634320066338112*x^2
> 0 - 4509318893445328263106912*x^18 + 6060765603004828811362460*x^16 + 159543
> 3811519440861448928*x^14 - 17237571860589604851458432*x^12 + 155539333684122
> 61315069024*x^10 + 11439096820765481332943688*x^8 - 248990211573927140987579
> 84*x^6 + 12932816995946524978942144*x^4 - 2199266291848578048793312*x^2 + 20
> 628044238087844473601
> )}
> 
> takes much longer than it should (IMO) before realizing the field is not Galois.

galoisinit() was designed on the premices that the input is a Galois
polynomial and does not go out of its way to catch non-Galois polynomial
sooner.

If you want a probabilistic check, use numberofconjugates:
install("numberofconjugates","lGD0,L,");
numberofconjugates(P)
%1 = 32
(So P has (at most) 32 automorphisms, and hence is not Galois).

In this specific case, galoisinit() is unlucky: none of the fourty
prime numbers it tried shows that P is not galois.

Cheers,
Bill.