Re: Fastest way to determine whether a simple extension is Galois, starting from a polynomial

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

On Tue, Jan 21, 2020 at 08:32:49PM +0100, Ricardo Buring wrote:
> Dear pari-users,
> 
> I notice that running e.g.
> 
>     length(nfgaloisconj(x^29 - 10*x + 13)) == 29
> 
> is much faster than
> 
>     length(nfgaloisconj(nfinit(x^29 - 10*x + 13))) == 29

Yes. nfinit computes the full factorization of the discriminant of the
polynomial, which is not needed.

> Are both of these valid ways to discover that Q[x]/(x^29 - 10*x + 13)
> is not Galois? 
Yes.

> And do these two tests give the same result general?
Yes.

> The documentation of nfgaloisconj only states that passing a
> polynomial is allowed; does it also mean that the answer to both
> questions is yes?
Yes.

Cheers,
Bill