Determines if $g$ is a primitive element of the Galois field $\mathrm{GF}(q^m)$ with degree-$m$ irreducible polynomial $f(x)$ over $\mathrm{GF}(q)$.

Parameters:¶

element: PolyLike¶

An element $g$ of $\mathrm{GF}(q^m)$ is a polynomial over $\mathrm{GF}(q)$ with degree less than $m$.

irreducible_poly: Poly¶

The degree-$m$ irreducible polynomial $f(x)$ over $\mathrm{GF}(q)$ that defines the extension field $\mathrm{GF}(q^m)$.

Returns:¶

True if $g$ is a primitive element of $\mathrm{GF}(q^m)$.

Examples¶

In the extension field $\mathrm{GF}(3^4)$, the element $x+2$ is a primitive element whose order is $3^4-1=80$.

In [1]: GF = galois.GF(3**4)

In [2]: f = GF.irreducible_poly; f
Out[2]: Poly(x^4 + 2x^3 + 2, GF(3))

In [3]: galois.is_primitive_element("x + 2", f)
Out[3]: True

In [4]: GF("x + 2").multiplicative_order()
Out[4]: 80

However, the element $x+1$ is not a primitive element, as noted by its order being only 20.

In [5]: galois.is_primitive_element("x + 1", f)
Out[5]: False

In [6]: GF("x + 1").multiplicative_order()
Out[6]: 20