galois.is_square_free¶
- galois.is_square_free(value)¶
Determines if the positive integer or the non-constant, monic polynomial is square-free.
- Parameters
value (int, galois.Poly) – A positive integer \(n\) or a non-constant, monic polynomial \(f(x)\).
- Returns
True
if the integer or polynomial is square-free.- Return type
Notes
A square-free integer \(n\) is divisible by no perfect squares. As a consequence, the prime factorization of a square-free integer \(n\) is
\[n = \prod_{i=1}^{k} p_i^{e_i} = \prod_{i=1}^{k} p_i .\]Similarly, a square-free polynomial \(f(x)\) has no irreducible factors with multiplicity greater than one. Therefore, its canonical factorization is
\[f(x) = \prod_{i=1}^{k} g_i(x)^{e_i} = \prod_{i=1}^{k} g_i(x) .\]Examples
Determine if an integer is square-free.
In [1]: galois.is_square_free(10) Out[1]: True In [2]: galois.is_square_free(16) Out[2]: False
Determine if a polynomial is square-free over \(\mathrm{GF}(3)\).
In [3]: GF = galois.GF(3) In [4]: g3 = galois.irreducible_poly(3, 3); g3 Out[4]: Poly(x^3 + 2x + 1, GF(3)) In [5]: g4 = galois.irreducible_poly(3, 4); g4 Out[5]: Poly(x^4 + x + 2, GF(3)) In [6]: galois.is_square_free(g3 * g4) Out[6]: True In [7]: galois.is_square_free(g3**2 * g4) Out[7]: False