Re: Galois test

[Karim Belabas <Karim.Belabas@math.u-psud.fr>]

* Ariel Pacetti [2005-06-17 13:56]:
> Is there a routine for checking wether a number field extension is Galois? 
> I couldn´t find one, but probably there is some "naive" way to do that 
> like:
> 
> nffactor(nfinit(P),P)
> 
> and check wether all the factors have degree one or not. Is there a better 
> (or faster) way? (like no using nfinit which takes too long if the 
> polynomial is big enough).

There's no built-in routine.  You may

-- check factorisation pattern mod a few primes first, which quickly
   weeds out (most) non-Galois fields.

-- use nfroots instead of nffactor (smaller bounds used).

-- possibly use factornf when you want to skip the 'nfinit' part.

-- still use nfinit _but_ read

     http://www.math.u-psud.fr/~belabas/pari/doc/faq.html#nfpartialfact

   first. In particular the following hack is often helpful:

        nfinitpartial(P) = nfinit( [P, nfbasis(P,1)] )

Hope this helps,

    Karim.
-- 
Karim Belabas                     Tel: (+33) (0)1 69 15 57 48
Dep. de Mathematiques, Bat. 425   Fax: (+33) (0)1 69 15 60 19
Universite Paris-Sud              http://www.math.u-psud.fr/~belabas/
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