class property galois.FieldArray.primitive_elements : FieldArray

All primitive elements \(\alpha\) of the Galois field \(\mathrm{GF}(p^m)\).

Notes

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

All primitive elements of the prime field \(\mathrm{GF}(31)\) in increasing order.

In [1]: GF = galois.GF(31)

In [2]: GF.primitive_elements
Out[2]: GF([ 3, 11, 12, 13, 17, 21, 22, 24], order=31)

In [3]: GF = galois.GF(31, repr="power")

In [4]: GF.primitive_elements
Out[4]: GF([   α, α^23, α^19, α^11,  α^7, α^29, α^17, α^13], order=31)

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

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

In [6]: GF.primitive_elements
Out[6]: GF([ 5,  9, 10, 13, 15, 17, 20, 21], order=5^2)

In [7]: GF = galois.GF(5**2, repr="poly")

In [8]: GF.primitive_elements
Out[8]: 
GF([     α,  α + 4,     2α, 2α + 3,     3α, 3α + 2,     4α, 4α + 1],
   order=5^2)

In [9]: GF = galois.GF(5**2, repr="power")

In [10]: GF.primitive_elements
Out[10]: GF([   α, α^17,  α^7, α^23, α^19, α^11, α^13,  α^5], order=5^2)