Factors the monic polynomial \(f(x)\) into a product of square-free polynomials.

Returns:¶

• The list of non-constant, square-free polynomials \(h_j(x)\) in the factorization.

• The list of corresponding multiplicities \(j\).

Raises:¶

ValueError – If \(f(x)\) is not monic or has degree 0.

Notes¶

The Square-Free Factorization algorithm factors \(f(x)\) into a product of \(m\) square-free polynomials \(h_j(x)\) with multiplicity \(j\).

Some \(h_j(x) = 1\), but those polynomials are not returned by this function.

A complete polynomial factorization is implemented in factors().

References¶

• Hachenberger, D. and Jungnickel, D. Topics in Galois Fields. Algorithm 6.1.7.

• Section 2.1 from https://people.csail.mit.edu/dmoshkov/courses/codes/poly-factorization.pdf

Examples¶

Suppose \(f(x) = x(x^3 + 2x + 4)(x^2 + 4x + 1)^3\) over \(\mathrm{GF}(5)\). Each polynomial \(x\), \(x^3 + 2x + 4\), and \(x^2 + 4x + 1\) are all irreducible over \(\mathrm{GF}(5)\).

In [1]: GF = galois.GF(5)

In [2]: a = galois.Poly([1,0], field=GF); a, a.is_irreducible()
Out[2]: (Poly(x, GF(5)), True)

In [3]: b = galois.Poly([1,0,2,4], field=GF); b, b.is_irreducible()
Out[3]: (Poly(x^3 + 2x + 4, GF(5)), True)

In [4]: c = galois.Poly([1,4,1], field=GF); c, c.is_irreducible()
Out[4]: (Poly(x^2 + 4x + 1, GF(5)), True)

In [5]: f = a * b * c**3; f
Out[5]: Poly(x^10 + 2x^9 + 3x^8 + x^7 + x^6 + 2x^5 + 3x^3 + 4x, GF(5))

The square-free factorization is \(\{x(x^3 + 2x + 4), x^2 + 4x + 1\}\) with multiplicities \(\{1, 3\}\).

In [6]: f.square_free_factors()
Out[6]: ([Poly(x^4 + 2x^2 + 4x, GF(5)), Poly(x^2 + 4x + 1, GF(5))], [1, 3])

In [7]: [a*b, c], [1, 3]
Out[7]: ([Poly(x^4 + 2x^2 + 4x, GF(5)), Poly(x^2 + 4x + 1, GF(5))], [1, 3])