Igor Schein on Fri, 17 Jun 2005 17:41:54 +0200

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 Re: Galois test

```On Fri, Jun 17, 2005 at 03:38:56PM +0200, Karim Belabas wrote:
> * 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,

I am using the following function:

isgalois(pol, gal) = if(polisirreducible(pol),if(!gal,gal=nfgaloisconj(pol,4));if(#Set(gal)==poldegree(pol),return(1)));return(0)

Igor

```

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