- classmethod galois.FieldArray.primitive_root_of_unity(n: int) Self
Finds a primitive \(n\)-th root of unity in the finite field.
- Parameters:¶
- Returns:¶
The primitive \(n\)-th root of unity, a 0-D scalar array.
- 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_root_of_unity(2) Out[2]: GF(30, order=31) In [3]: GF.primitive_root_of_unity(5) Out[3]: GF(16, order=31) In [4]: GF.primitive_root_of_unity(15) Out[4]: GF(9, order=31)
However, they do not exist for \(n\) that do not divide 30.
In [5]: GF.primitive_root_of_unity(7) --------------------------------------------------------------------------- ValueError Traceback (most recent call last) Cell In[5], line 1 ----> 1 GF.primitive_root_of_unity(7) File ~/checkouts/readthedocs.org/user_builds/galois/envs/stable/lib/python3.8/site-packages/galois/_fields/_array.py:890, in FieldArray.primitive_root_of_unity(cls, n) 888 raise ValueError(f"Argument 'n' must be in [1, {cls.order}), not {n}.") 889 if not (cls.order - 1) % n == 0: --> 890 raise ValueError(f"There are no primitive {n}-th roots of unity in {cls.name}.") 892 return cls.primitive_element ** ((cls.order - 1) // n) 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_root_of_unity(5); root Out[6]: GF(16, order=31) In [7]: powers = np.arange(1, 5 + 1); powers Out[7]: array([1, 2, 3, 4, 5]) In [8]: root ** powers Out[8]: GF([16, 8, 4, 2, 1], order=31)