Computer program for factorization into irreducible polynomials over $\mathbb{Z}_{p^k}$

Question

Hensel's Lemma allows us to factor a polynomial uniquely into basic irreducible factors over $\mathbb{Z}_{p^k}$. Is there a SAGE or Magma command that gives this factorization? Or can anyone help in writing a small script that handles this problem? Thanks in advance.

Answer 1

For example, you can do this with GAP, using Factors:

gap> f:= CyclotomicPolynomial( GF(2), 7 );
x_1^6+x_1^5+x_1^4+x_1^3+x_1^2+x_1+Z(2)^0
gap> Factors( f );
[ x_1^3+x_1+Z(2)^0, x_1^3+x_1^2+Z(2)^0 ]
gap> Factors( PolynomialRing( GF(8) ), f );
[ x_1+Z(2^3), x_1+Z(2^3)^2, x_1+Z(2^3)^3, x_1+Z(2^3)^4, 
  x_1+Z(2^3)^5, x_1+Z(2^3)^6 ]

(see GAP manual here for the documentation of Factors).