galois.FieldArray.field_norm

galois.FieldArray.field_norm() → FieldArray

Computes the field norm \(\mathrm{N}_{L / K}(x)\) of the elements of \(x\).

Returns:¶: The field norm of \(x\) in the prime subfield \(\mathrm{GF}(p)\).

Notes¶

The self array \(x\) is over the extension field \(L = \mathrm{GF}(p^m)\). The field norm of \(x\) is over the subfield \(K = \mathrm{GF}(p)\). In other words, \(\mathrm{N}_{L / K}(x) : L \rightarrow K\).

For finite fields, since \(L\) is a Galois extension of \(K\), the field norm of \(x\) is defined as a product of the Galois conjugates of \(x\).

\[\mathrm{N}_{L / K}(x) = \prod_{i=0}^{m-1} x^{p^i} = x^{(p^m - 1) / (p - 1)}\]

References¶

https://en.wikipedia.org/wiki/Field_norm

Examples¶

Compute the field norm of the elements of \(\mathrm{GF}(3^2)\).

Integer Polynomial Power

In [1]: GF = galois.GF(3**2)

In [2]: x = GF.elements; x
Out[2]: GF([0, 1, 2, 3, 4, 5, 6, 7, 8], order=3^2)

In [3]: y = x.field_norm(); y
Out[3]: GF([0, 1, 1, 2, 1, 2, 2, 2, 1], order=3)

In [4]: GF = galois.GF(3**2, repr="poly")

In [5]: x = GF.elements; x
Out[5]: 
GF([     0,      1,      2,      α,  α + 1,  α + 2,     2α, 2α + 1,
    2α + 2], order=3^2)

In [6]: y = x.field_norm(); y
Out[6]: GF([0, 1, 1, 2, 1, 2, 2, 2, 1], order=3)

In [7]: GF = galois.GF(3**2, repr="power")

In [8]: x = GF.elements; x
Out[8]: GF([  0,   1, α^4,   α, α^2, α^7, α^5, α^3, α^6], order=3^2)

In [9]: y = x.field_norm(); y
Out[9]: GF([0, 1, 1, 2, 1, 2, 2, 2, 1], order=3)