[GAP Forum] galois group

[Alexander Hulpke]

Dear GAP Forum,

Marta Aseda wrote:
>
> I am wondering if galois group computation using gap is useable as a
> mathematical result.

In general, unless you explicitly turn off verifications or use  
functions which deliberately only return proibabilistic results (such  
as `ProbabilityShapes' for Galois groups), any result obtained with  
(documented functions of) GAP is proven correct. (There is of course  
always the philosophical problem of human error in  implementation or  
use, however the same problems are as well in published papers.) The  
aim is that one could use such a result in the same way as a result  
from a published paper.

> Is there any ambiguity ? Or, if I give a polynomial,
> and it computes its galois group, is it mathematically certain that  
> the
> polynomial I gave was irreducible ? How does gap know that it is  
> irreducible

The command `GaloisType' first performs an irreducibility test, using  
the standard approach of factorization modulo a prime, Hensel lifting  
and testing systematically all combinations of the lifted factors.  
(There is a newer algorithm due to van Hoeij, but this is not yet  
implemented in GAP.)
You can check this directly by calling `Factors' on the polynomial.
Thus indeed it is certain that -- if no error is issued -- the  
polynomial is irreducible. Also the type returned by `GaloisType' is  
proven correct -- at the cost that such a test might take very long  
for certain polynomials.

Best wishes,

    Alexander Hulpke