Factoring polynomials over finite fields using Maple and Maxima

Question

Can you factor polynomials over certain finite fields using Maple or Maxima, and if so, how?

Answer 1

In Maple, you can use Factor and mod, e.g. for factoring over $GF(7)$

Factor(x^2 + 3 x + 3) mod 7;

$$(x+6)(x+4)$$

For $GF(p^k)$, represented using a root of a polynomial of degree $k$ irreducible over $GF(p)$:

alias(alpha = RootOf(x^3 + 2));

Factor(x^4 + x^2 + 2*x + 2, alpha) mod 7;

$$(5 \alpha^2 + x + 1)(x+4)(6 \alpha^2 + x + 1)(3 \alpha^2 + x + 1)$$