Primitive n-th Root of Unity - Explanation

We call \(\zeta\) an \(n\)-th root of unity if \(\zeta^n=1\), and we call \(\zeta\) a primitive \(n\)-th root of unity if, in addition, \(\zeta^k\neq 1\) for \(1\leq k\leq n-1\). What is the purpose of this additional requirement?

Consider an example \(\zeta:=-1\) satisfies

\[\zeta^4 = (-1)^4 = 1\]

So \(\zeta\) is a 4th root of unity. However, we also have

\[\zeta^2 = (-1)^2 = 1\]

So \(\zeta=-1\) is not a primitive 4th root of unity. It is, however, a primitive second root of unity. So this characterizes what is special about a primitive nth root of unity: \(n\) is the least positive integer with \(\zeta^n=1\).

For example, \(\omega_3:=\cos\tfrac{2\pi}{3} + \sin\tfrac{2\pi}{3}\cdot i\) and \(\omega_{3}^{2}\) are both primitive 3rd roots of unity.

For example, if \(\omega_{12} = \cos\tfrac{2\pi}{12} + \sin\tfrac{2\pi}{12}\cdot i\), then the primitive 12th roots of unity are

\[\omega_{12},\ \omega_{12}^{5},\ \omega_{12}^{7},\ \omega_{12}^{11}\]

On the other hand

• \(\omega_{12}^{2}\) is not a primitive 12th root of unity because \[\left( \omega_{12}^{2}\right)^{6} = \omega_{12}^{12} = 1\]
• \(\omega_{12}^{3}\) is not a primitive 12th root of unity because \[\left( \omega_{12}^{3}\right)^{4} = \omega_{12}^{12} = 1\]
• \(\omega_{12}^{4}\) is not a primitive 12th root of unity because \[\left( \omega_{12}^{4}\right)^{3} = \omega_{12}^{12} = 1\]
• \(\omega_{12}^{6}\) is not a primitive 12th root of unity because \[\left( \omega_{12}^{6}\right)^{2} = \omega_{12}^{12} = 1\]
• \(\omega_{12}^{8}\) is not a primitive 12th root of unity because \[\left( \omega_{12}^{8}\right)^{3} = \omega_{12}^{24} = 1^2 = 1\]
• \(\omega_{12}^{9}\) is not a primitive 12th root of unity because \[\left( \omega_{12}^{9}\right)^{4} = \omega_{12}^{36} = 1^4 = 1\]
• \(\omega_{12}^{10}\) is not a primitive 12th root of unity because \[\left( \omega_{12}^{10}\right)^{6} = \omega_{12}^{60} = 1^5 = 1\]

If you are comfortable with concepts from the theory of groups, we can give the following view of primitive roots of unity: if \(\omega_n=\cos\tfrac{2\pi}{12} + \sin\tfrac{2\pi}{12}\cdot i\), the nth roots of unity

\[1,\ \omega_n,\ \omega_{n}^{2},\ \cdots,\ \omega_{n}^{n-1}\]

form a cyclic group of order \(n\). The primitive roots nth roots of unity are exactly the generators of the group. They are given by those powers \(\omega_{n}^{k}\) for which the greatest common divisor of \(k\) and \(n\) is \(1\). On the other hand, whenever \(k\) and \(n\) have a common divisor that is greater than \(1\), then \(\omega_{n}^{k}\) is a root of unity, but not a primitive one.