classmethod galois.FieldArray.primitive_roots_of_unity(n: int) → Self

Finds all primitive \(n\)-th roots of unity in the finite field.

Parameters:
n: int

The root of unity.

Returns:

All primitive \(n\)-th roots of unity, a 1-D array. The roots are sorted in lexicographical order.

Raises:

ValueError – If no primitive \(n\)-th roots of unity exist. This happens when \(n\) is not a divisor of \(p^m - 1\).

Notes

A primitive \(n\)-th root of unity \(\omega_n\) is such that \(\omega_n^n = 1\) and \(\omega_n^k \ne 1\) for all \(1 \le k \lt n\).

In \(\mathrm{GF}(p^m)\), a primitive \(n\)-th root of unity exists when \(n\) divides \(p^m - 1\). Then, the primitive root is \(\omega_n = \alpha^{(p^m - 1)/n}\) where \(\alpha\) is a primitive element of the field.

Examples

In \(\mathrm{GF}(31)\), primitive roots exist for all divisors of 30.

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

In [2]: GF.primitive_roots_of_unity(2)
Out[2]: GF([30], order=31)

In [3]: GF.primitive_roots_of_unity(5)
Out[3]: GF([ 2,  4,  8, 16], order=31)

In [4]: GF.primitive_roots_of_unity(15)
Out[4]: GF([ 7,  9, 10, 14, 18, 19, 20, 28], order=31)

However, they do not exist for \(n\) that do not divide 30.

In [5]: GF.primitive_roots_of_unity(7)
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
Cell In[5], line 1
----> 1 GF.primitive_roots_of_unity(7)

File /opt/hostedtoolcache/Python/3.11.10/x64/lib/python3.11/site-packages/galois/_fields/_array.py:952, in FieldArray.primitive_roots_of_unity(cls, n)
    950     raise TypeError(f"Argument 'n' must be an int, not {type(n)!r}.")
    951 if not (cls.order - 1) % n == 0:
--> 952     raise ValueError(f"There are no primitive {n}-th roots of unity in {cls.name}.")
    954 roots = np.unique(cls.primitive_elements ** ((cls.order - 1) // n))
    955 roots = np.sort(roots)

ValueError: There are no primitive 7-th roots of unity in GF(31).

For \(\omega_5\), one can see that \(\omega_5^5 = 1\) and \(\omega_5^k \ne 1\) for \(1 \le k \lt 5\).

In [6]: root = GF.primitive_roots_of_unity(5); root
Out[6]: GF([ 2,  4,  8, 16], order=31)

In [7]: powers = np.arange(1, 5 + 1); powers
Out[7]: array([1, 2, 3, 4, 5])

In [8]: np.power.outer(root, powers)
Out[8]: 
GF([[ 2,  4,  8, 16,  1],
    [ 4, 16,  2,  8,  1],
    [ 8,  2, 16,  4,  1],
    [16,  8,  4,  2,  1]], order=31)