Re: the minimal polynomial over the composite field

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

On Wed, Oct 09, 2019 at 02:33:31PM +0200, Bill Allombert wrote:
> On Wed, Oct 09, 2019 at 12:46:15PM +0900, macsyma wrote:
> > Factoring by galoisfixedfield(,,2) is attractive, 
> > but it seems time-consuming to put together preparations for that.
> 
> This is true. galoissplittinginit is doing much more initialization
> that what you need. You need much smaller bound.

In particular: galoisfixedfield(G,2) returns three components
[subf, inc, fact] where subf is the subfield generator, inc the
inclusion from subf to G.pol and fact the factorisation of
G.pol over subf.

Of this, the most costly to compute is inc because it is an element
of the large field. In anticipation of this galoissplittinginit
compute the inverse of the van der Monde matrix.

If you do not need inc, then you need a version on galoissplittinginit
that does not compute the inverse and a version of galoisfixedfield
that does not compute inc.

(Note: when I wrote galoisfixedfield, galoissplittinginit did not exist,
so the only way was to use galoisinit which require the inverse of the
van der Monde matrix anyway )

galoisspittinginit is part of a new project which try to reconciliate galoisinit and
polgalois. Your feedback is very useful.

Cheers,
Bill.