 In this video, we're going to show that not every regular polygon can be constructed using a straight edge and compass alone. Now, in order to do that, to solve this last unsolvable geometric construction, we need to provide a little bit of context. Someone tells a little bit of history. So going back to the 19th century, the so-called Prince of Mathematics, Carl Gauss, was the first to consider the constructability of real numbers and its relationship to algebra, which is what this whole lecture 28 has been about in our lecture series. The relationship is due to the solutions to the cyclotonic equation x to the n minus 1 equals 0. This is an equation we have studied previously back in Math 4220 in the previous lecture series, for which the solutions to this are in fact going to be the complex roots of unity, the complex nth roots of unity, I should say. Those are the n roots of this thing. And so Gauss was the first to show that this equation is always solvable by radicals. The roots of unity are always solvable by radicals. He also showed that if n is a prime number and p minus 1 is not a power of 2, then the solutions involve radicals higher than degree 2. And the solutions of the cyclotonic polynomial and the complex numbers, they coincide with the roots of unity, you already mentioned that. And so now we introduce the idea of a Fermat prime. So a Fermat prime is a prime of the following form. It's one greater than a power of 2. Although you can argue that the power of 2 is itself a power of 2. I won't dive into that too much right now. But by the previous result that I was talking about right here, when it comes to these roots of unity, so say zeta is a root of unity, e to the 2 pi i over n, like so. Zeta n is a root to the polynomial x to the n minus 1. Now, this is not the minimal polynomial. That gets a little bit more complicated. We'll talk about that some other time in this lecture series. But what one can prove is that if you take q adjoin a primitive nth root of unity and look at this as a field extension over q, then this is always equal to phi of n, where phi of n here is Euler's totient function. This counts the number of integers less than n, which are co-prime to n. When p is a prime number, this number is going to be p minus 1. All right. And when you have a prime power, of course, phi of p to the n, this looks like p to the n minus p to the n minus 1, like so. When your numbers are co-prime, a times b, phi then becomes phi of a times phi of b. Again, this happens if the GCD of a and b equals 1. And so using these two observations, how to compute totient of a prime power and totient of co-prime numbers, you can compute the totient of any integer so long as you know how to factor that thing. And so it comes down to the constructability of the regular ngon, comes down to this phi function. Because of this right here, if you can construct the regular ngon, it means you can construct this complex number. Now, this is a complex number, right? So we're thinking of it, of course, as cosine of 2 pi over n plus i sine of 2 pi over n. And so this is a point in the plane. So the real part is the x-coordinate and the imaginary part is the y-coordinate. If we can construct cosine of 2 pi over n, we can construct sine of 2 pi over the n. So it comes down to can we construct cosine of 2 pi over n? And now as we were considering trisecting an angle, we discovered that there were some cosines that we couldn't do. Like, we couldn't do cosine of 20 degrees. Is that a possibility coming on right here? So this conversation of Fermat primes comes into play because a Fermat prime has the property that phi of p. So you take 2 to the 2k plus 1 here. Since it's a prime, this is going to be 2 to the 2k, like so. So for phi of a Fermat prime, it's a power of 2, all right? Now, this is true for any Fermat prime. Now, if the Fermat prime is repeated because of the property we mentioned earlier, you no longer get a power of 2. You get a power of 2 times something else, okay? Now, if you take a power of 2 itself, 2 to the k, this is going to be 2 to the k minus 2 to the k minus 1 for which this then becomes 2 to the k minus 1. You can simplify it in the following way. So if you take phi of 2 to a power, that's a power of 2. If you take phi of a Fermat prime, that's a power of 2. Therefore, if you have a product of a power of 2 and any number of Fermat primes, which that could be no Fermat primes, then phi of that number will be a power of 2. And those are the only integers so that their Euler-Toschent is a power of 2, okay? And so remember with the constructable numbers. With the constructable numbers here, if you have a constructable number alpha, if you look at the degree of q of joint alpha over q, this is always equal to a power of 2. But when you look at the complex roots of unity, you can only construct them with a degree phi of n in that situation. And therefore, if this is a power of 2, this only happens in the situation where you have a product of 2s times some number of distinct Fermat primes. Now, Gauss was made famous because he was the first one to discover that you could construct using a compass and a straight edge, the regular 17 gone, which 17 is a Fermat prime. Notice the Fermat primes. You have 3, which is 2 plus 1. You have the next one, 5, which is 4 plus 1. Notice here, we're just taking a power of 2 plus 1. The next one, a plus 1, it's not prime, that's 9. Then you take 16 plus 1, that's 17. That's a Fermat prime. So Gauss was able to prove that you could construct the regular 7 gone. Now, the shapes that we can do like a triangle, oh, triangle 3 is a Fermat prime. We can do that. You can construct a square. Well, a square is 2 square. That's a power of 2. You can construct a pentagon because 5 is a Fermat prime. You can construct a hexagon, a regular hexagon here, because a hexagon 6 is 2 times 3. That's a power of 2 times a Fermat prime. You could also construct 12, a 12-sided figure, a dodecagon, because 12 is 4 times 3. We could construct a 20 gone, because 20 is 4 times 5, like so. But 17, of course, was this really cool one that Gauss was able to produce. As legend goes, Gauss had requested that the regular 17 gone be engraven upon his tombstone, because he was so impressed by this result. Now, the engraver of the tombstone, again, as legend, refused to do so because he claimed that the engraving was too complicated to do. So you can look at Gauss's tombstone, Google the picture of it. You don't see the regular 17 gone there, and that's really too bad here. So we're returning to the mathematics here. If we were to construct the regular in gone, we need to be able to basically, we need phi of n to be a power of 2, because this boils down to be able to construct cosine of 2 pi over n, which note here that cosine of 2 pi over n, this is the same thing as 1 half, a primitive complex root of unity plus its conjugate, which is also the same thing as its inverse, right? So cosine of 2 pi over n, it belongs to this field, okay? And so that's why this degree does matter. And cosine of 2 pi over n, if you look at the degree of the extension, q adjoined a primitive root, a primitive nth root over q adjoined cosine of 2 pi over n. This degree is, this is degree two extension because cosine of 2 pi over n is the root of this quadratic polynomial, x squared minus zeta plus zeta inverse x plus one, which is a polynomial over this field, okay? For which case then, if you look at q adjoined cosine of 2 pi over n as an extension over q, then the degree of that extension is phi of n over 2. So the constructability comes down to is phi of n over 2, a power of 2. And the conditions that Gauss had mentioned earlier, exactly that. Your number n needs to look like some power of 2 and you have some products of distinct Fermat primes, which again, that product could be empty. Now I should make mention that with, at the recording of this video, there are only five known Fermat primes. We've already mentioned three of them, three, five and 17. There's only five known. Fermat actually conjectured that every number in that sequence was gonna be a prime number. He basically only computed the first four. I think he could did the first four. Maybe he did all five of them. I don't think so, but he constructed the first four and then conjectured that they were always prime. The very next one was not prime, whoopsie daisy. So Fermat has a bad habit of getting things named after him even though he never proves anything. In this case, his conjecture was actually wrong. Fermat numbers are not always Fermat primes. There's five known Fermat primes. And honestly, the current mathematical community believes there are no more Fermat primes that there are only five of them, but the proof has yet to be provided. And so it's appropriate to end this conversation about impossibilities with yet an open theorem. How many regular n-gons can you construct? Because we don't know how many Fermat primes there are. So we actually don't know how many n-gons we can construct. We know it has to have this factorization, but how many primes are there? How many Fermat primes? It's probably the five, but you know, someone hopefully one day will prove that result too.