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

Thank you very much.

________________________________________
From: Alexander Hulpke [hul...@fastmail.fm]
Sent: 05 June 2015 03:28 PM
To: Jaco Versfeld
Cc: fo...@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.versf...@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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