galois.GF2.normal_elements

property galois.GF2.normal_elements : FieldArray

All normal elements 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¶

All normal elements of the extension field \(\mathrm{GF}(5^2)\) in lexicographical order.

Integer Polynomial Power

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

In [2]: GF.normal_elements
Out[2]: 
[GF(5, order=5^2),
 GF(6, order=5^2),
 GF(8, order=5^2),
 GF(9, order=5^2),
 GF(10, order=5^2),
 GF(11, order=5^2),
 GF(12, order=5^2),
 GF(13, order=5^2),
 GF(15, order=5^2),
 GF(17, order=5^2),
 GF(18, order=5^2),
 GF(19, order=5^2),
 GF(20, order=5^2),
 GF(21, order=5^2),
 GF(22, order=5^2),
 GF(24, order=5^2)]

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

In [4]: GF.normal_elements
Out[4]: 
[GF(α, order=5^2),
 GF(α + 1, order=5^2),
 GF(α + 3, order=5^2),
 GF(α + 4, order=5^2),
 GF(2α, order=5^2),
 GF(2α + 1, order=5^2),
 GF(2α + 2, order=5^2),
 GF(2α + 3, order=5^2),
 GF(3α, order=5^2),
 GF(3α + 2, order=5^2),
 GF(3α + 3, order=5^2),
 GF(3α + 4, order=5^2),
 GF(4α, order=5^2),
 GF(4α + 1, order=5^2),
 GF(4α + 2, order=5^2),
 GF(4α + 4, order=5^2)]

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

In [6]: GF.normal_elements
Out[6]: 
[GF(α, order=5^2),
 GF(α^22, order=5^2),
 GF(α^2, order=5^2),
 GF(α^17, order=5^2),
 GF(α^7, order=5^2),
 GF(α^8, order=5^2),
 GF(α^4, order=5^2),
 GF(α^23, order=5^2),
 GF(α^19, order=5^2),
 GF(α^11, order=5^2),
 GF(α^16, order=5^2),
 GF(α^20, order=5^2),
 GF(α^13, order=5^2),
 GF(α^5, order=5^2),
 GF(α^14, order=5^2),
 GF(α^10, order=5^2)]