Primitive roots of unity

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

Consider the first nth 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 nth 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 nth roots of 1?

Of course, the answer is yes! These dots are called the primitive nth 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 nth root of unity is primitive if it is not also an mth 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 6th 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.

I will leave to you the fun of figuring out why these definitions are the same!

About Brent

Associate Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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