Primitive roots of unity

So we have now seen that there are always $n$ different complex $n$ th roots of unity, that is, complex numbers whose $n$ th power is equal to $1$ , equally spaced around the circumference of the unit circle.

Consider the first $n$ th root around the circle from the positive $x$ -axis (i.e. the darkest blue dot in the picture above). Let’s call this number $\omega_n$ . For example, in a previous post we saw that $\omega_6 = 1/2 + i \sqrt{3}/2$ . Then the other dots around the circle—the other $n$ th roots of 1—are obtained by taking powers of $\omega_n$ . The next dot is $\omega_n^2$ , the next is $\omega_n^3$ , … and finally $\omega_n^n = 1$ .

In my original post that kicked off this series, I drew circles with dots on only some of the spokes, like this:

I explained one way to think about this: in the circle with $n$ spokes, there is a dot on spoke $k$ (where the spoke on the positive $x$ -axis is numbered 0) if and only if $k$ and $n$ are relatively prime, that is, $k$ and $n$ have no common divisors other than $1$ ( $\gcd(k,n) = 1$ ). Now that we know how to think of the dots as complex numbers, we can also ask: what is special about these dots as complex numbers? Is there a way to rephrase this definition in terms of $n$ th roots of 1?

Of course, the answer is yes! These dots are called the primitive $n$ th roots of unity. In the above picture with $n=12$ , using our new notation, the highlighted dots are $\omega_{12}$ , $\omega_{12}^5$ , $\omega_{12}^7$ , and $\omega_{12}^{11}$ . In general, we will have $\omega_n^k$ where $\gcd(k,n) = 1$ . But there is a different, equivalent way to characterize them: an $n$ th root of unity is primitive if it is not also an $m$ th root of unity for some smaller $m$ . For example, consider $\omega_{12}^2$ , corresponding to the second spoke on the circle above. This is a 12th root of 1, but it is not a primitive 12th root of 1, because it is also a $6$ th root of 1. (In fact, it is our old friend $\omega_6 = 1/2 + i \sqrt{3}/2$ .) In other words, if we start with $\omega_{12}^2$ and take successive powers of it, we find a power (the sixth power) that yields 1 before we get to the 12th power. Something similar is true for $\omega_{12}^3$ : it is a 12th root of 1, sure, but raising it to the 12th power is overkill—just raising it to the 4th power will get us to 1. $\omega_{12}^5$ , on the other hand, is a primitive 12th root—we actually have to multiply it by itself 12 times before reaching $1$ .