[GAP Forum] Factorizing polynomials over GF(2^m)?

[Alexander Hulpke]

Dear GAP Forum,

> On Jun 5, 2015, at 7:19 AM, Jaco Versfeld <Jaco.Versfeld at wits.ac.za> wrote:
> 
> I want to factor polynomials over GF(2^m).  As a quick test, I did the following:
> 
> R:=PolynomialRing(GF(8),["x"]);
> x:=Indeterminate(GF(8),"x");
> p := x^7 + 1;
> Factors(p);
> 
> The result that I obtain is:
> [ x+Z(2)^0, x^3+x+Z(2)^0, x^3+x^2+Z(2)^0 ]
> 
> This doesn't make sense, since I expected (x-\alpha^0), (x-\alpha^1) ... (x-\alpha^6) to have been the roots.

Polynomials do not carry the actual ring, but only the characteristic and get factored over their coefficient rings. To factor over GF(8), specify the polynomial ring, i.e.
gap> Factors(R,p);  
[ x+Z(2)^0, x+Z(2^3), x+Z(2^3)^2, x+Z(2^3)^3, x+Z(2^3)^4, x+Z(2^3)^5, x+Z(2^3)^6 ]

Regards,

   Alexander Hulpke