 In this video, we're gonna prove the very important property of unique factorization domains, which recall a unique factorization domain is an integral domain with unique factorization. That is, if an element has a factorization, well, every element other than units and zero has a factorization into irreducibles and those irreducibles are in fact unique up to reordering and association. That is to say, every non-unit non-zero element has a quote unquote prime factorization. Why do we call it a prime factorization as opposed to an irreducible factorization? Well, that's because in a unique factorization domain every irreducible element is a prime element. Now, I want to remind you that in a previous video we proved that every prime element in an integral domain is an irreducible element. In a unique factorization domain, this is reversible. Now, be aware, we've seen also previously that in a domain, you can have an irreducible that's not prime, but in a unique factorization domain the two elements are equivalent. The two notions of prime and irreducible is the same. So that's why in unique factorization domains we call this unique factorization irreducibles the prime factorization. This is why kids in primary school when they learn about prime numbers they define them as irreducible numbers but they call them prime numbers. Well, that's because the ring of integers is a UFD. So primes and irreducibles are the same thing. So no harm, no foul in that setting. So let's see the proof of this thing. So we're gonna take an element X inside of the UFD and suppose it is irreducible. So every factorization of X involves a unit to some degree. So now remember what does it mean to be a prime? To be a prime means you satisfy Euclid's lemma. If you divide, if your number divides a product then it must divide one of the factors. That's what we want to argue right now. So suppose X divides a product A times B. Well, since X divides A times B that means there exists some element Y inside the domain so that A times B is equal to X times Y. This is where unique factorizations come to play here because we have two different factorizations of the same number. So let's look at those factorizations. Now the number A has a unique factorization. There's some product of irreducible elements PI and since we're commuting an associative you can group them together. So let epsilon I be its exponent so we just gather the same primes together, okay? And so we then get something like this. I should say irreducibles. We haven't yet proven that they're prime. So A has some product in some irreducible product. B likewise has some product in irreducibles and these prime factorizations are unique up to reordering an association of course. So then the factorization of A times B is then gonna be take all the prime factor all the irreducible factors of A get all the irreducible factors of B put them together. And so this is a prime factorization of AB since we're in a UFD this factorization is unique, okay? So you have to compare that then with the prime factorization of X times Y because we're in a unique factorization domain these have to be the same but X itself by assumption is an irreducible. So one of the irreducible factors that shows up in the prime factorization of X times Y is X itself. So by uniqueness of factorization since X is irreducible, X must be an associate to either PI or QJ for some I or some J. So without the loss of generality let's assume that U is a unit so that U times X is equal to PI. So X and PI are associates inside of this prime factorization. Well, what does this mean then? So this means that X divides PI but PI was a factor of A divisibility is transitive. So if X divides PI and PI divides A that means that X time or X divides A, right? And we could pull all this things out, right? You know, we could write, you know, we could factor this a little bit more. We basically are gonna substitute in in the factorization of A where did it go? Factorization of A, you know, we can get rid of the P one. Let's say that this was equal to P one. We can replace P one with this. And so X is in there and you can move the unit somewhere else. So we actually have X, X does in fact divide A. And so since X divided A that shows it is a prime element. So in a unique factorization domain, prime elements and irreducible elements are exactly the same thing.