 Welcome back to our lecture series, Math 42-30, Abstract Algebra 2 for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. In lecture 32 here, we are going to continue our discussion of polynomial codes leading up to what we call a BCH code, which we'll define that in the last video for lecture 32. In the meanwhile though, in preparation to define BCH codes, we have to talk about roots of unity, for which in the prequel Math 42-20, Abstract Algebra 1, to this lecture series, we did introduce complex roots of unity, which would be the roots of unity over the rational field, and basically every other field of, well, every other field contained inside the complex numbers, we'll put it that way. We actually are going to take an opportunity right now to extend the notion of a root of unity, also in particular a primitive root of unity, we want to define a root of unity over any field. This will include finite fields, which will then have applications to our coding theory here. So let F be a field. This could be a finite field, it could be an infinite field, it could be characteristic zero, characteristic P. It doesn't matter, just take any field, and we'll say that zeta is some element of that field. We say that zeta is an nth root of unity, if zeta to the n equals one. Another way of saying that is that if you take the polynomial f of x to equal x to the n minus one, and you view this as an f polynomial here, then f evaluated at zeta is then equal to zero. So we say that zeta is a root of unity, if it's a root of this polynomial x to the n minus one, which of course is equivalent to saying that the nth power of zeta gives you one. Now, much like we did before with complex numbers, we say that zeta is a primitive root of unity, if no smaller power accomplishes this. So for example, if you take zeta to be a primitive sixth root of unity, then we're gonna see that of course zeta squared is also a sixth root, because after all, if I take zeta squared and I raise that to the sixth power, you're gonna end up with what x one of the laws, whoops, this is the same thing as zeta to the sixth squared, for which of course zeta to the sixth is one. So you end up with one squared, which is then one. So zeta squared is also a sixth root of unity, because zeta was a sixth root of unity, like we have right here. But of course, zeta squared is not a primitive sixth root because while the sixth power does give you one, notice instead that if we take zeta squared and we take that to the third power, this will actually end up, again, by exponent laws, this is the same thing as zeta to the two times three power, zeta six, so this is equal to one. So the third power of zeta six will get you there. And in fact, one can argue that zeta squared here, if zeta was a sixth root of unity, zeta squared will actually be a primitive third root of unity. And so these are relations we saw previously when we talked about complex roots of unity. So next, I wanna introduce the symbol phi sub n of x and this is a polynomial in f of joint x. And this will denote the minimal polynomial of zeta over f and we call this the nth cyclotonic polynomial. And admittedly, over the rational field, there is one and only one cyclotonic polynomial for each root of unity class n, like so. And in that situation, over the rational numbers, the degree of this cyclotonic polynomial phi of n is actually equal to phi of n, where phi in this case is actually Euler's totient function, the function that computes the number of integers co-prime that are less than n, all right? And that's actually why we use this capital phi to denote it in that situation. Now over other fields, these things can look very, very differently. In fact, over the rational field, this polynomial phi of n contains as its roots, all of the primitive nth roots of unity. But over other fields, they can actually split into more than one class. That is, you have some primitive roots over here, some other primitive roots that have the same degree. And so it gets a little bit more complicated, but still we'll refer to this as the nth cyclotonic polynomial. Now, I should mention, why are we talking about this right now? Let's note that roots of unity are actually very important in field theory. In particular, their important in our discussion of cyclic codes. That'll be the basis upon these roots of unity will be where we're gonna define our BCH codes from. So let F be a finite field and if Zeta is an nth root of unity, you're still gonna have this relationship Zeta n equals one. This is true for every field, right? Whether you're finite or infinite doesn't really matter. This statement is gonna hold true right there. But in the case of a finite field, I want you to remember that if F here is finite, then F star, remember, is going to equal. Let's say that this is order n right here. This is going to be isomorphic to the cyclic group Zn minus one, right? And so this right here says something about the order of Zeta inside of F star. If we think of it, F star is this cyclic group. Well, this tells you that the order of Zeta is gonna divide n. Now, if it's a primitive nth root of unity, then the order of Zeta, multiplicatively speaking, is exactly n. But by Lagrange's theorem, it's gonna divide that. And so that's like we saw earlier how a third root of unity is a sixth root of unity. And so as we study this cyclic group, we see that F star contains, it definitely contains an element of order n, if and only if we get that n divides the order here. So let me clean this up a little bit. Typically when we come to these finite fields, we think of it as P to the K, of course. So that means F star as a cyclic group is gonna be F to the P to the K minus one. And so if Zeta is a primitive nth root of unity, then that means that n must divide P to the K minus one. So how you factor this power of a prime tells you a lot. So about the field. And so if the order of F is P to the K, that field will contain nth root of unity if and only if n divides P to the K minus one. And so we might be interested in is F a splitting field for the polynomial X the n minus one or not. So the splitting field for this polynomial X the n minus one is gonna be the smallest power of K such that n divides P to the K minus one. And so that gives us the splitting field. So for example, let me slide up a little bit more so I can write this out here. So for example, if we look at the polynomial X to the 15th minus one, this splits over F 16, of course. Clearly 16 minus one is 15, 15 divides 15. And so you're gonna get the X to the 15 minus one splits over F 16 right here. And we can also argue that this is in fact the smallest field finite field of characters to two for which X to the 15 minus one splits, right? Because if you look at the other ones like F eight that's it's way too small, 15 is bigger than eight same thing with four or two as well. So this is in fact the splitting field for this polynomial. This tells us that F 16 contains all of the 15th roots of unity, which include of course the primitive 15th roots of unity. We also get the primitive fifth roots, the primitive third roots and then of course the primitive first root as well. And we're actually gonna build a BCH code using the factorization of X to the 15th minus one. Again, we'll do this at the end of this video of course. So over a field for which X to the N minus one does not split, notice that X to the N minus one will factor into irreducible factors of course because our polynomial rings are still gonna be unique factorization domains. So like for example, F to the 15 minus one splits over F 16 but what if we look at the field F two Z two? It's not gonna split over that field because again it doesn't contain the primitive 15th roots of unity. So it'll factor into some collection of irreducibles. So if P of X is an irreducible factor of X to the N minus one then one of its roots is a primitive Dth root of unity for some divisor of N. So note that necessarily the other roots of P of X must be other primitive Dth roots of unity but doesn't necessarily have to be all of them. So when we factor this thing into its irreducible factors whether it's X to the 15 minus one or in general X to the N minus one when we factor it in particular when we factor it over Z two which is the prime field for all of these characteristic two finite fields. When we factor it here it's gonna give you a bunch of irreducible factors and the roots of those factors are going to be in throats of unity. And let's say that when you have one of those factors here you have some P of X times some other things, right? You look at P of X, its roots are going to be in throats of unity and let's say that they're primitive Dth roots of unity where D is a divisor of N. All the other roots inside of P will also have to be primitive Dth roots of unity which will be in throats of unity not necessarily primitive though but be aware unlike the rationals it doesn't have to contain all of their roots. This is something we're gonna see in the future. Now speaking of irreducible polynomials and roots this is an appropriate time to bring up a very useful proposition. Suppose that F is a finite field of order P to the K so clearly it's characteristic P, P is the prime here and suppose we have some polynomial F of X inside of the polynomial field the polynomial ring should say F of joint X and let E be the splitting field for this polynomial F of X. So E contains all of the roots of F of X and let's say one of those roots is omega. So if omega is a root of F of X then it turns out that omega to the P to the K power is likewise a root. That is if you take omega and then raise it to the order of the finite field F that gives you another root of the same polynomial. And so this can be particularly useful when your base field is just the prime field. So you're looking at for example, ZP. This tells you that if omega is a root of your irreducible polynomial then omega to the P is also a root and then that to the P power and then that to the P power. Eventually this process will cycle around but this potentially could grab you other roots of the polynomial and potentially it could give you all of the other roots of the polynomial. And so that'll be very, very nice. We're of course gonna prove it in the more general case where F is a finite field of order P to the K but our primary emphasis will be when we have the prime field ZP right there. So okay, let's look at our polynomial F of X. Let's say the coefficients are C0, C1, C2 up to Cn, right? So our polynomial looks like C0 plus C1x plus C2x squared all the way up to Cn x to the n like so. And so then if we evaluate this at our root omega for which again, omega is just any number in, it could be any number right in these finite fields. It doesn't necessarily have to be a root of unity although of course roots of unity will be our ultimate focus as we develop this theory for BCH codes. So you look at the polynomial F of omega in which case evaluation just means you plug in omega for each of the X's, right? We get that. And since it's a root, this evaluation should equal zero, fantastic. So we have this equation right here. Consider the polynomial evaluated at omega P to the K. Well then evaluation looks something like the following. Now previously we have seen that with finite fields that if you raise an element of a finite field to the order of that finite field this actually acts like the identity on that element, right? So Fermat's little theorem is often represented over the finite field ZP but this works for any finite field whatsoever that if you have A belonging to a finite field if you take A and raise it to the order of that finite field, this in fact gives you back A inside of that field. So raising an element to the order of the field is like the identity of that field. So as each of these coefficients C0, C1 up to Cn as all of these belong to the field F, if I raise them to the order of F which is P to the K that won't do anything to them. So C naught is the same thing as C naught to the PK. C1 is the same thing as C1 to the PK but as you then have C1 to the PK times omega to the PK we can factor that as C1 times omega to the PK and then you continue on like so, all right? So then as we are working over a field of characteristic P what we can then do is we can actually use freshman exponentiation on these things as long as we're taking powers of P which is exactly what we're doing. So this will then turn into C naught plus C1 omega plus going all the way down we're gonna have a Cn omega to the N right here all raised to the PK power. We're allowed to do that because we're working mod P in this finite field but this right here is just F of omega. Like so we just have F omega to the PK power. Well, since F omega is zero we have then zero to the PK power and any finite power of zero is gonna likewise be zero as well. So this then establishes the fact that if you have a root of a polynomial that means any over a finite field then the root raised to the order of that base field for which the coefficients are coming from for the polynomial will also give you another root and this will be very, very, very helpful for us. So for example, if we look at the cyclic tonic polynomial we'll say Z phi 15 of X here and we're looking for this polynomial well clearly one of its roots we're looking at the splitting field right now one of this roots is going to be Zeta where Zeta is a primitive 15th root of unity and then by this proposition here since we're working mod two so we're thinking this over the field Z two here that means Z squared will be another root and then you're gonna have Z to the fourth is another root and then you're gonna have Z to the eighth which is another root. Then of course when you take Z 16 because these are primitive 15th roots of unity Z 16 is the same thing as Zeta 16 it's the same thing as Zeta 15 times Zeta to the first Zeta to the 15th is one so this just gives you back to Zeta. So as you cycle through this you're gonna get that Zeta is a root so is Zeta squared so is Zeta to the fourth so is Zeta to the eighth and then it cycles back are there any other roots? Well, we'll need another argument but sure enough we can prove that phi 15 has degree four over Z squared and thus this accommodates for all of the roots. So we can find a complete factorization utilizing this proposition with of course if we know the degree of the polynomial but we'll talk about that a little bit more next time.