Re: nfgaloisconj

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

On Tue, Jul 30, 2019 at 10:44:45AM +0900, macsyma wrote:
> Thank you, Bill.
> 
> > what it does
> 
> In my code, G1 is a permutation representation, G2 is a polynomial
> representation of G the Galois group of f over Q. The principle is
> directly linked to Q-automorphism, that for each m_j in G, the
> permutation of the roots of f is obtained by replacing alpha the
> primitive element in the root representation of f with m_j(alpha) the
> image of alpha that is a root of g. One can consider alpha =
> polroots(g)[1], m_j(alpha) = polroots(g)[i] in G12 code. 

This seems useful. Maybe we should add this algorithm to PARI (by using
p-adic roots).

> G12 can be applied even if G is not weakly super-solvable. However,
> for example, the processing time of
> https://www.math.u-bordeaux.fr/~ballombe/polynomials.gp takes 20 to 30
> times that of galoisinit + nfgaloisconj, so I'm hoping that you make
> speed up nfisincl and nfsplitting.

Note that polynomials.gp has been superceded by the galpol database:
<http://pari.math.u-bordeaux.fr/galpol/>.

It will be difficult to improve nfisincl and nfsplitting. 
This is a much harder problem than galoisinit.

Cheers,
Bill.