A primitive element \(\alpha\) of the Galois field \(\mathrm{GF}(p^m)\). A primitive element is a multiplicative generator of the field, such that \(\mathrm{GF}(p^m) = \{0, 1, \alpha, \alpha^2, \dots, \alpha^{p^m - 2}\}\).

A primitive element is a root of the primitive polynomial \(f(x)\), such that \(f(\alpha) = 0\) over \(\mathrm{GF}(p^m)\).

Examples¶

In [1]: galois.GF(2).primitive_element
Out[1]: GF(1, order=2)

In [2]: galois.GF(2**8).primitive_element
Out[2]: GF(2, order=2^8)

In [3]: galois.GF(31).primitive_element
Out[3]: GF(3, order=31)

In [4]: galois.GF(7**5).primitive_element
Out[4]: GF(7, order=7^5)