# galois.primitive_element¶

galois.primitive_element(irreducible_poly, start=None, stop=None, reverse=False)

Finds the smallest primitive element $$g(x)$$ of the Galois field $$\mathrm{GF}(p^m)$$ with degree-$$m$$ irreducible polynomial $$f(x)$$ over $$\mathrm{GF}(p)$$.

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

• start (int, optional) – Starting value (inclusive, integer representation of the polynomial) in the search for a primitive element $$g(x)$$ of $$\mathrm{GF}(p^m)$$. The default is None which represents $$p$$, which corresponds to $$g(x) = x$$ over $$\mathrm{GF}(p)$$.

• stop (int, optional) – Stopping value (exclusive, integer representation of the polynomial) in the search for a primitive element $$g(x)$$ of $$\mathrm{GF}(p^m)$$. The default is None which represents $$p^m$$, which corresponds to $$g(x) = x^m$$ over $$\mathrm{GF}(p)$$.

• reverse (bool, optional) – Search for a primitive element in reverse order, i.e. find the largest primitive element first. Default is False.

Returns

A primitive element of $$\mathrm{GF}(p^m)$$ with irreducible polynomial $$f(x)$$. The primitive element $$g(x)$$ is a polynomial over $$\mathrm{GF}(p)$$ with degree less than $$m$$.

Return type

galois.Poly

Examples

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

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

In [3]: galois.is_irreducible(f)
Out[3]: True

In [4]: galois.is_primitive(f)
Out[4]: True

In [5]: galois.primitive_element(f)
Out[5]: Poly(x, GF(3))

In [6]: GF = galois.GF(3)

In [7]: f = galois.Poly([1,0,1], field=GF); f
Out[7]: Poly(x^2 + 1, GF(3))

In [8]: galois.is_irreducible(f)
Out[8]: True

In [9]: galois.is_primitive(f)
Out[9]: False

In [10]: galois.primitive_element(f)
Out[10]: Poly(x + 1, GF(3))