galois.irreducible_poly
- galois.irreducible_poly(order, degree, method='min')
Returns a monic irreducible polynomial \(f(x)\) over \(\mathrm{GF}(q)\) with degree \(m\).
- Parameters
order (int) – The prime power order \(q\) of the field \(\mathrm{GF}(q)\) that the polynomial is over.
degree (int) – The degree \(m\) of the desired irreducible polynomial.
method (str, optional) –
The search method for finding the irreducible polynomial.
"min"
(default): Returns the lexicographically-minimal monic irreducible polynomial."max"
: Returns the lexicographically-maximal monic irreducible polynomial."random"
: Returns a randomly generated degree-\(m\) monic irreducible polynomial.
- Returns
The degree-\(m\) monic irreducible polynomial over \(\mathrm{GF}(q)\).
- Return type
Notes
If \(f(x)\) is an irreducible polynomial over \(\mathrm{GF}(q)\) and \(a \in \mathrm{GF}(q) \backslash \{0\}\), then \(a \cdot f(x)\) is also irreducible. In addition to other applications, \(f(x)\) produces the field extension \(\mathrm{GF}(q^m)\) of \(\mathrm{GF}(q)\).
Examples
The lexicographically-minimal, monic irreducible polynomial over \(\mathrm{GF}(7)\) with degree \(5\).
In [1]: p = galois.irreducible_poly(7, 5); p Out[1]: Poly(x^5 + x + 3, GF(7)) In [2]: galois.is_irreducible(p) Out[2]: True
Irreducible polynomials scaled by non-zero field elements are also irreducible.
In [3]: GF = galois.GF(7) In [4]: galois.is_irreducible(p * GF(3)) Out[4]: True
A random, monic irreducible polynomial over \(\mathrm{GF}(7^2)\) with degree \(3\).
In [5]: p = galois.irreducible_poly(7**2, 3, method="random"); p Out[5]: Poly(x^3 + 2x^2 + 3x + 17, GF(7^2)) In [6]: galois.is_irreducible(p) Out[6]: True