galois.is_irreducible

class galois.is_irreducible(poly)

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

A polynomial \(f(x) \in \mathrm{GF}(p)[x]\) is reducible over \(\mathrm{GF}(p)\) if it can be represented as \(f(x) = g(x) h(x)\) for some \(g(x), h(x) \in \mathrm{GF}(p)[x]\) 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-\(n\) polynomial \(f(x)\) over \(\mathrm{GF}(p)\) for prime \(p\) is irreducible if and only if \(f(x)\ |\ (x^{p^n} - x)\) and \(\textrm{gcd}(f(x),\ x^{p^{m_i}} - x) = 1\) for \(1 \le i \le k\), where \(m_i = n/p_i\) for the \(k\) prime divisors \(p_i\) of \(n\).

Parameters

poly (galois.Poly) – A polynomial \(f(x)\) over \(\mathrm{GF}(p)\).

Returns

True if the polynomial is irreducible.

Return type

bool

References

Examples

# Conway polynomials are always irreducible (and primitive)
In [502]: f = galois.conway_poly(2, 5); f
Out[502]: Poly(x^5 + x^2 + 1, GF(2))

# f(x) has no roots in GF(2), a requirement of being irreducible
In [503]: f.roots()
Out[503]: GF([], order=2)

In [504]: galois.is_irreducible(f)
Out[504]: True

In [505]: g = galois.conway_poly(2, 4); g
Out[505]: Poly(x^4 + x + 1, GF(2))

In [506]: h = galois.conway_poly(2, 5); h
Out[506]: Poly(x^5 + x^2 + 1, GF(2))

In [507]: f = g * h; f
Out[507]: Poly(x^9 + x^5 + x^4 + x^3 + x^2 + x + 1, GF(2))

# Even though f(x) has no roots in GF(2), it is still reducible
In [508]: f.roots()
Out[508]: GF([], order=2)

In [509]: galois.is_irreducible(f)
Out[509]: False