 Let zeta be a non-zero complex number and let n be a positive integer, you know, one, two, three, four, whatever. We say that zeta is an nth root of unity if zeta to the n is equal to one. So any power, any complex number raised to this power n, we call, if that power equals one, we say it's an nth root of unity. So some examples of this, we have previously seen that the numbers one negative one i and negative i are fourth roots of unity. Notice that one to the fourth is equal to one. Negative one to the fourth is gonna equal one because after all negative one squared is one and you square one, you're gonna get one. We also have seen that i to the fourth is equal to one. And likewise, if you take negative i to the fourth, this is the same thing as negative one to the fourth times i to the fourth, which is gonna be one times one, which is one. So these are fourth roots of unity. We say that a root of unity is primitive. It's a primitive nth root of unity. If no smaller power of n will get you to get you one, right? So what I mean here is that n is the smallest positive integer that makes zeta to the n equal to one. So for example, when you look at these numbers right here, one, you could actually get one by much smaller power. Notice if you take one to the first, that will equal one. So one is not, although it's a fourth root of unity, it's not a primitive fourth root of unity. And then if you look at negative one also, negative one, we could, as I mentioned earlier, if you take negative one squared, that's equal to one. And so again, negative one's not a primitive fourth root of unity. It is a fourth root of unity because the fourth power gives you one, but it's not primitive. On the other hand, when you look at i and negative i, these are in fact primitive nth roots of unity, because no smaller power of i that's positive will get you one sooner than four. Same thing when you take negative i right here. Now conversely, if you take the set one, negative one, this consisted of the set of square roots of unity or second roots of unity. One squared is one, negative one squared is one. Now as we saw, negative one is a primitive, and so when we talk about primitive, it matters on the degree here. So i, negative i are primitive fourth roots. Negative one is a primitive second root of unity. And then one would be a primitive first root of unity. I mean, it's actually the only first root of unity, but it's a primitive first root. And so these are some examples of, these are examples of roots of unity that you're probably familiar with. What we wanna do is actually expand this list and talk about roots of unity in general. Now in order to do that, we have to introduce something called Euler's Identity, which the proof of that basically uses, it basically uses power series from calculus for which there'll be a review video at the end of this video that you can click on to review those topics if you need. And so in order to kind of really give justice to what Euler's Identity is all about, we have to introduce the geometry of complex numbers. Now what we can do is we can actually associate points in the plane with complex numbers. So what I mean is something like the following. If we were to graph our usual x, y axes and we identify the real axis as the x-axis and we identify the y-axis with the imaginary axis, then points in the plane can be identified, these x, y can be identified with complex numbers x plus y, i. And there is a geometry to that. This essentially leads to the polar form of a complex number. So every number, every complex number z can be written as r times e to the i theta, where r is called the modulus of the complex number, that's a y, and theta is the argument of the complex number, which the argument's gonna be this angle, the line forms with the real axis, and then the distance the point is from the origin, we call the modulus of the complex number. So every complex number z can be written in this form, r times e to the i theta, where e to the i theta is actually equal to cosine of theta plus i sine theta. And so one can then do some calculations with this if one wanted to. Again, look at some of the videos you're gonna see at the end of this one. If you want some more reference on this, this is gonna be a really critical thing for us as we do computations with these complex numbers. Let me show you two immediate consequences of Euler's identity. Euler's identity basically tells us that if you wanna do multiplication with complex numbers, you're better using the polar form than the standard Cartesian form, that is z equals x plus yi. This is the standard slash Cartesian form of the complex number. You wanna use polar forms when you multiply together complex numbers. So let's say we have two complex numbers, say z is equal to r times cosine theta plus i sine theta and w is equal to s times cosine theta plus i sine theta, and that's not a theta, I'm sorry, s cosine phi plus i sine of phi. So you multiply together the two complex numbers, z times w. This will actually equal the product of their moduli and then you actually add together the argument. So add together theta and phi right there. And so this actually proves that the modulus, kind of like the absolute value, has this multiplicative property. The modulus of z w is equal to the modulus of z times the modulus of w. Sometimes this is also called the norm of a complex number, all three of those terms, modulus, norm, and absolute value are used to describe this quantity for complex numbers. I wanna mention that this formula right here is an immediate consequence of Euler's identity because if we take z times w, this is gonna equal r e to the i theta times s e to the i phi. And therefore, if you multiply these things together, you're going to end up with r s times e to the i theta plus i phi by usual exponent rules, and it follows immediately from there. The next thing to mention here, and this is gonna be, this is the critical thing about these complex roots of unity, this is actually gonna be the key to how we can build complex roots of unity. Let's introduce a new symbol. We're gonna call zeta sub n. It's gonna equal this complex number, e to the pi i over n, two pi i over n. So you can think of this as e to the i times two pi over n. Now by Euler's identity, this thing is just the same thing as cosine of two pi over n plus i sine two pi over n. Now we claim that this number zeta n is a primitive nth root of unity. Well, why is that? Let's first argue why it's a nth root of unity in the first place. This basically follows from a very classic identity that's a consequence of Euler's identity right here. z to the n is gonna equal r to the n times cosine of n theta plus i sine of n theta. This is a consequence of the previous corollary, right? Exponents in polar form are very easy when it comes to complex numbers. So if we apply this to zeta to the n, if you raise that to the n, you're gonna end up with e to the two pi, oops, two pi i to the n to the n. By exponent rules, this will give you e to the two pi i, which by Euler's identity would then be cosine of two pi plus i sine of two pi. Sine of two pi is zero, cosine of two pi is one. So we see that this is an nth root of unity. I claim it's also a primitive nth root of unity that there's no smaller power that can get this to work. And you can check by induction that any smaller power, I guess it's not even induction arguments necessary here, that you can try that for a smaller n, some smaller value than n you're not going to get, you're not gonna get one, not at all. And also I wanna mention that if you take s into the k, where k is co-prime to n, by similar reasoning, this is going to be a primitive nth root of unity because the only way that you're gonna get cosine to be one and sine to be zero is if you're at a multiple of two pi, that's the only way to do it. And because of the divisor n, the denominator n, I should say here, there's no way you can cancel out the denominator until you get up to n itself. And that's even if you have something co-prime because if you're co-prime, k and n have no common divisors. So there's no way of simplifying the fraction earlier than at the nth step. So we can construct primitive nth roots of unity over the complex numbers for any, any n we want. Now, if some of this went too fast, again, there's some review videos I want you to take a look at right now if you need some. Looking at the top left, you can see a video for review on the complex plane. How do we graph complex numbers and visualize them as points in the plane? To the top right, you can see a video link for the modulus, the argument, and the polar form of a complex number. Take a look at that if you need to. In the bottom left, you'll see a video that provides the proof of Euler's identity. And also it gives you some examples how one does computations using Euler's identity, how we can evaluate things like e to the pi i is equal to negative one and such. So click on those links if you need any review on these things whatsoever. If you feel ready to go, click the link in the bottom right, which then we'll get to the main point of this lecture for which we can then construct cyclic groups inside of C star using these roots of unity.