Polynomial factorization over finite fields with non-prime order

Asked Jul 11 '13 at 12:32

Active Jul 18 '13 at 20:55

Viewed 1,279 times

One can easily factor a polynomial over finite fields of prime order, using Factor command:

Factor[1 + x^2, Modulus -> 2]
===>  (1 + x)^2

Now, is it possible to do this over finite fields of non-prime order? As an example, $ x^2 + x + 1$ is reducible over GF(4), and can be decomposed as $(x-a)(x-a^2)$, where $a\not\in \{0,1\} $ is a field member.

Problem is, members of non-prime order finite fields are not denoted by ordinary numbers; rather, they use either polynomial or vector notation.

edited Jul 18 '13 at 20:55

asked Jul 11 '13 at 12:32

Sadeq Dousti

See http://reference.wolfram.com/mathematica/FiniteFields/tutorial/FiniteFields.html It says Factor will not work with the Finite Fields package. – Michael E2 Jul 18 '13 at 23:54
@MichaelE2: So, Mathematica offers no way to factor over finite fields with non-prime order? – Sadeq Dousti Jul 18 '13 at 23:56
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No built-in functions that I know of. Others might. You could implement an algorithm or you could perhaps use MathLink to call PARI from within Mathematica: http://pari.math.u-bordeaux.fr/dochtml/mathlink.html See also this related question and the answer. – Michael E2 Jul 19 '13 at 00:10
@MichaelE2: Thanks for introducing me to MathLink. I used GUAVA to do math over finite fields, but I'm gonna take a look at PARI/GP, as it can be called from Mathematica using MathLink. – Sadeq Dousti Jul 19 '13 at 09:53
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You may be able to call a GAP script. See GAP FAQ, Run, etc., this comment, the discussion of this question. – Michael E2 Jul 19 '13 at 13:15
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One way to go about this would be to implement the Cantor-Zassenhaus algorithm over finite extensions to prime fields. – Daniel Lichtblau Jul 19 '13 at 22:40

Polynomial factorization over finite fields with non-prime order

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