galois.FieldArray.field_norm

galois.FieldArray.field_norm() → FieldArray

Computes the field norm $N_{L / K} (x)$ of the elements of $x$ .

Returns:¶: The field norm of $x$ in the prime subfield $GF (p)$ .

Notes

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

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 $x$ .

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 $GF (3^{2})$ .

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)