 In this video, we're gonna show that every Euclidean domain is in fact a principal ideal domain. So what we have to do is using the Euclidean norm, demonstrate that an arbitrary ideal is in fact a principal ideal. So clearly if I take the zero ideal, that's just generated by the zero element itself. So we don't have to worry about that. So let I be a non-trivial ideal of this Euclidean domain. I will call it D for the sake of it. And so let B be some non-zero element inside of the ideal. And we're gonna choose it so it has the smallest norm. Because of the Euclidean norm that every Euclidean domain has, we have this function, call it new, that will send our elements to something inside the natural numbers. As it's the set of natural numbers, we have the well-ordering principle. So there is an element of smallest norm. We're gonna choose that element to be B. This is actually why we threw zero out. Because by construction, the norm is not defined for the zero element. So since B is inside of the ideal, the ideal itself, I, contains the principal ideal generated by B. And so what we wanna argue is that this ideal is in fact equal to the principal ideal generated by B. That's gonna be our plan of attack here. So we have to argue that every other element the ideal is a power of B. I should say it's a product of B of some kind. So take some other element A. Now I don't have to rule out the possibility of being zero here because zero times B would be zero, no big deal. So take an element A that's inside of this ideal. Since we have a Euclidean domain, we have the division algorithm. That is given A and given B, there's gonna exist a unique quotient and remainder such that A equals QB plus R, which we know that R is either itself the zero element or the norm of R is strictly less than the norm of B. Now since I is an ideal that contains B, the product QB must belong to the ideal. I mean, it's an ideal after all. And so if you manipulate this equation, this is the same thing as saying R is equal to A minus QB for which A is inside the ideal by assumption, QB is inside the ideal since B is in the ideal and well, it's an ideal. Therefore, the difference R minus QB is inside of the ideal. That difference is this remainder R, okay? Now we had two possibilities on the remainder R. It either was zero, in which case no big deal. Zero belongs to the ideal. But the other possibility is if R is non-zero, then the norm of R is gonna be less than the norm of B. Since R belongs to the ideal, this then would contradict the minimality of our choice of B, because right, B had minimum norm. This R can't have a smaller norm. The only escape hatch possible is that R was equal to zero. Now if R equals zero, that would mean that this equation, since it's equal to zero, we would get that A equals QB, which is an element of the principle ideal generated by B, as we can see right here. And since A was an arbitrary element, this would then show that I is contained as a subset inside the principle ideal, thus giving us the equality we wanted. So every ideal in the Euclidean domain is in fact a principle ideal. That then proves it's a principle ideal domain. Now as we've seen previously, every principle ideal domain is a unique factorization domain. So every Euclidean domain is likewise a unique factorization domain. In a Euclidean domain, we have unique factorization, unique prime factorizations. It's pretty nice. And this comes from a consequence of this division algorithm, for which again, we're modeling this based upon the behavior we saw in the integer ring because of the division algorithm, because of the well-ordered principle, the ring of integers has a lot of properties. And in fact, every Euclidean domain will have similar properties as well. So Euclidean domain is both a PID and a UFD because every PID is a UFD. We have seen some examples in previous lectures that an integral domain doesn't have to be a UFD. We've also, well, we've talked about examples where a UFD doesn't have to be a principle ideal domain. We'll prove that some other time. I did wanna close this video with stating that it is possible for a principle ideal domain to not be a Euclidean domain, that these are not equivalent notions. And so I want you to consider the element theta, which we're gonna define that to be one plus the square root of negative 19 over two. And then I want you to consider the integers joined this element theta. So it takes a little bit of effort to prove it. I'm not gonna provide the details in this video, but this is an example of a principle ideal domain. One can show that there is an argument there, but it's also not a Euclidean domain. And this is probably the simplest example, but itself gets actually kind of complicated, which is why I'm skipping it over from this video right here. These types of connections and these types of observations are very important in algebraic number theory. And so I will defer someone who wants the details to either produce them yourself to look them up. But it's a bit of a technical argument that we will take for granted right now.