Determines whether the polynomial \(f(x)\) over \(\mathrm{GF}(p^m)\) is irreducible.

Returns:¶

True if the polynomial is irreducible.

Notes¶

A polynomial \(f(x) \in \mathrm{GF}(p^m)[x]\) is reducible over \(\mathrm{GF}(p^m)\) if it can be represented as \(f(x) = g(x) h(x)\) for some \(g(x), h(x) \in \mathrm{GF}(p^m)[x]\) of strictly lower degree. If \(f(x)\) is not reducible, it is said to be irreducible. Since Galois fields are not algebraically closed, such irreducible polynomials exist.

This function implements Rabin’s irreducibility test. It says a degree-\(m\) polynomial \(f(x)\) over \(\mathrm{GF}(q)\) for prime power \(q\) is irreducible if and only if \(f(x) \mid (x^{q^m} - x)\) and \(\textrm{gcd}(f(x),\ x^{q^{m_i}} - x) = 1\) for \(1 \le i \le k\), where \(m_i = m/p_i\) for the \(k\) prime divisors \(p_i\) of \(m\).

References¶

• Rabin, M. Probabilistic algorithms in finite fields. SIAM Journal on Computing (1980), 273-280. https://apps.dtic.mil/sti/pdfs/ADA078416.pdf

• Gao, S. and Panarino, D. Tests and constructions of irreducible polynomials over finite fields. https://www.math.clemson.edu/~sgao/papers/GP97a.pdf

• Section 4.5.1 from https://cacr.uwaterloo.ca/hac/about/chap4.pdf

• https://en.wikipedia.org/wiki/Factorization_of_polynomials_over_finite_fields

Examples¶

# Conway polynomials are always irreducible (and primitive)
In [1]: f = galois.conway_poly(2, 5); f
Out[1]: Poly(x^5 + x^2 + 1, GF(2))

# f(x) has no roots in GF(2), a necessary but not sufficient condition of being irreducible
In [2]: f.roots()
Out[2]: GF([], order=2)

In [3]: f.is_irreducible()
Out[3]: True

In [4]: g = galois.irreducible_poly(2**4, 2, method="random"); g
Out[4]: Poly(x^2 + 13x + 10, GF(2^4))

In [5]: h = galois.irreducible_poly(2**4, 3, method="random"); h
Out[5]: Poly(x^3 + 7x^2 + 5x + 14, GF(2^4))

In [6]: f = g * h; f
Out[6]: Poly(x^5 + 10x^4 + 10x^3 + x^2 + 14x + 6, GF(2^4))

In [7]: f.is_irreducible()
Out[7]: False