property galois.GF2.normal_element : FieldArray | None

A normal element of the finite field \(\mathrm{GF}(p^m)\).

Notes

A normal element \(\beta\) is one whose Frobenius conjugates

\[ \{\beta, \beta^p, \beta^{p^2}, \dots, \beta^{p^{m-1}}\} \]

form a basis of \(\mathrm{GF}(p^m)\) over its base field \(\mathrm{GF}(p)\). Prime fields (\(m = 1\)) have no nontrivial normal elements.

See also

galois.is_normal_element

Examples

The smallest normal element of the extension field \(\mathrm{GF}(5^2)\).

In [1]: GF = galois.GF(5**2)

In [2]: GF.normal_element
Out[2]: GF(5, order=5^2)

In [3]: GF = galois.GF(5**2, repr="poly")

In [4]: GF.normal_element
Out[4]: GF(α, order=5^2)

In [5]: GF = galois.GF(5**2, repr="power")

In [6]: GF.normal_element
Out[6]: GF(α, order=5^2)