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

[Jaco Versfeld]

Thank you very much.

________________________________________
From: Alexander Hulpke [hulpke at fastmail.fm]
Sent: 05 June 2015 03:28 PM
To: Jaco Versfeld
Cc: forum at gap-system.org
Subject: Re: [GAP Forum] Factorizing polynomials over GF(2^m)?

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


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