Computes the additive order of each element in $$x$$.

Returns:

An integer array of the additive order of each element in $$x$$. The return value is a single integer if the input array $$x$$ is a scalar.

Notes

The additive order $$a$$ of $$x$$ in $$\mathrm{GF}(p^m)$$ is the smallest integer $$a$$ such that $$x a = 0$$. With the exception of 0, the additive order of every element is the finite field’s characteristic.

Examples

Compute the additive order of each element of $$\mathrm{GF}(3^2)$$.

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]: order = x.additive_order(); order
Out[3]: array([1, 3, 3, 3, 3, 3, 3, 3, 3])

In [4]: x * order
Out[4]: GF([0, 0, 0, 0, 0, 0, 0, 0, 0], order=3^2)

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

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

In [7]: order = x.additive_order(); order
Out[7]: array([1, 3, 3, 3, 3, 3, 3, 3, 3])

In [8]: x * order
Out[8]: GF([0, 0, 0, 0, 0, 0, 0, 0, 0], order=3^2)

In [9]: GF = galois.GF(3**2, repr="power")

In [10]: x = GF.elements; x
Out[10]: GF([  0,   1, α^4,   α, α^2, α^7, α^5, α^3, α^6], order=3^2)

In [11]: order = x.additive_order(); order
Out[11]: array([1, 3, 3, 3, 3, 3, 3, 3, 3])

In [12]: x * order
Out[12]: GF([0, 0, 0, 0, 0, 0, 0, 0, 0], order=3^2)