6.11 afactors (17.3.99)

6.11.1 Werner Heiss

The command

evala(AFactors(-t^7-t^3-2*t^6-2*t^2+2))

didn’t terminate after several hours. Is it normal that it takes so long or might it be a bug?

6.11.2 Michael Monagan (18.3.99)

Yes, you are asking it to compute the splitting field which may be degree 7! = 5040 – it will have to factor a polynomial of degree 5040 over Z.

The computation blows up. Even degree 5 is non trivial in the general case.

6.11.3 Nils Bruin (19.3.99)

This polynomial is the product of a linear factor and an irreducible degree 6 factor. The degree 6 factor has galois group S6, so you’re asking maple to construct a degree 6! extension (as a degree 2 ext. of a degree 3ext etc.) This is a huge task, so I’d consider it normal that this takes so long. I can imagine that the answer wouldn’t even fit in your computer.

6.11.4 Helmut Kahovec (19.3.99)

IMO there is not a bug here: your polynomial is just too complicated for being decomposed into linear factors by Maple. Look at the following (simple) example:

> restart; 
> alias(alpha=RootOf(Z^6+Z^5+Z^4+Z^3+Z^2+Z+1)); 
 
> expr1:=t^7-1; 
 
> evala(AFactor(expr1)); 
 
                                4 
  (t - 1) (t - alpha) (t - alpha ) 
 
                              2        3        4        5 
        (t + 1 + alpha + alpha  + alpha  + alpha  + alpha ) 
 
                  2            3            5 
        (t - alpha ) (t - alpha ) (t - alpha )
 

Anyway, your polynomial may in fact be partially decomposed into linear factors:

> -t^7-2*t^6-t^3-2*t^2+2; 
 
                        7      6    3      2 
                      -t  - 2 t  - t  - 2 t  + 2 
 
> factor(%); 
 
                           6    5    4    3 
                -(1 + t) (t  + t  - t  + t  + 2 t - 2) 
 
> expr2:=remove(type,%,{linear,constant}); 
 
                           6    5    4    3 
                 expr2 := t  + t  - t  + t  + 2 t - 2 
 
> alias(beta=RootOf(Z^6+Z^5-Z^4+Z^3+2*Z-2)); 
 
> map(collect,factor(expr2,beta),[t]); 
 
       4           2       3       5 
  (beta  + 2 + beta  - beta  + beta 
 
                4              2       3 
         + (beta  + beta - beta  + beta ) t 
 
                3                  2   2               4 
         + (beta  + 1 - beta + beta ) t  + (1 + beta) t 
 
                           2   3    5 
         + (beta - 1 + beta ) t  + t ) (t - beta)
 

Apparently, the nonlinear part of this is very complicated. Maple might not be able to decompose it any further.