galois.is_primitive_element

galois.is_primitive_element(element, irreducible_poly)

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

The number of primitive elements of \(\mathrm{GF}(p^m)\) is \(\phi(p^m - 1)\), where \(\phi(n)\) is the Euler totient function, see galois.euler_totient.

Parameters

• element (galois.Poly) – An element \(g(x)\) of \(\mathrm{GF}(p^m)\) as a polynomial over \(\mathrm{GF}(p)\) with degree less than \(m\).

• irreducible_poly (galois.Poly) – The degree-\(m\) irreducible polynomial \(f(x)\) over \(\mathrm{GF}(p)\) that defines the extension field \(\mathrm{GF}(p^m)\).

Returns

True if \(g(x)\) is a primitive element of \(\mathrm{GF}(p^m)\) with irreducible polynomial \(f(x)\).

Return type

bool

Examples

In [389]: GF = galois.GF(3)

In [390]: f = galois.Poly([1,1,2], field=GF); f
Out[390]: Poly(x^2 + x + 2, GF(3))

In [391]: galois.is_irreducible(f)
Out[391]: True

In [392]: galois.is_primitive(f)
Out[392]: True

In [393]: g = galois.Poly.Identity(GF); g
Out[393]: Poly(x, GF(3))

In [394]: galois.is_primitive_element(g, f)
Out[394]: True

In [395]: GF = galois.GF(3)

In [396]: f = galois.Poly([1,0,1], field=GF); f
Out[396]: Poly(x^2 + 1, GF(3))

In [397]: galois.is_irreducible(f)
Out[397]: True

In [398]: galois.is_primitive(f)
Out[398]: False

In [399]: g = galois.Poly.Identity(GF); g
Out[399]: Poly(x, GF(3))

In [400]: galois.is_primitive_element(g, f)
Out[400]: False