 So in this video I want to find the notion of a Euclidean norm sometimes called a Euclidean function or Euclidean valuation map and With that we can then define what an Euclidean domain is. So first of all, let D be an integral domain So it's a commutative ring with unity and we have no proper divisors of zero We have the cancellation axiom just the usual definition of integral domain and then define New to be a norm that is it's a map from the non-zero elements of the domain to the natural numbers where zero is included in that Remember to be a norm we have to have the property that the norm of x is less than or equal to the norm of x y Where y is any non-zero element inside of the domain as well We say that a norm is a Euclidean norm Again, there's some other vocabulary people can use here We'll call it just a Euclidean norm in this situation A norm is Euclidean if for all elements a and b inside of the Non-zero elements of the domain there exist elements q and r which they could possibly be zero such that a equals qb plus r and in particular this this element r is itself either zero or It's a it has a norm that is strictly less than the norm of b now when you look at these things together this looks a lot like the Division algorithm we saw for integers which I haven't even finished the definition yet But the integers are very much the poster child of what a Euclidean domain is because a Euclidean domain is going to be any integral domain which we have a Division algorithm which of course let me finish the definition there a Domain d that's equipped with a Euclidean norm is known as a Euclidean domain And so essentially in a Euclidean domain, where does it get the name Euclidean a Euclidean domain will be a domain For which the Euclidean algorithm exists. We can actually compute the gcd between any two Elements using the division algorithm for which we've then have abstract What does that division algorithm means it means the following where we have this equation a equals qb plus r Whereas usual q you can think of as the quotient r is the remainder and the remainder should get smaller Either it's zero notice that in the definition of a norm. We don't actually define the norm of zero So the norm of the excuse me the remainder is either zero or the norm got strictly smaller And so this guarantees that when we do things like division or the Euclidean algorithm that the process will ultimately Terminate at some point or another So again Euclidean domains get their names because of the Euclidean algorithm And so we're trying to generalize the notion that we see of course with the integers now Another example worth noting here of course is the Gaussian integers that if we take the norm that we've been Experimenting with about the Gaussian integers that is the norm of a Gaussian integers just its absolute value squared You take the complex number times it by its conjugate you get a square plus b squared This is likewise a Euclidean norm and therefore you can use this norm to perform Division on Gaussian integers for finding GCDs some things that we'll talk about of course and just a little bit here likewise The ring Z join the square root of 2 it is an example of Euclidean domain for which its Euclidean norm Would be the following you're going to take the product of a plus b square root of 2 Multiply it by its conjugate a minus b times square root of 2 then you take its you take its absolute value That is you're going to get a real number a squared minus 2b squared And you take that absolute value whether it's positive negative it'll take its absolute value this gives us that Z you join the square root of 2 is likewise a Euclidean domain I'm going to leave it up to the viewer to prove the details of some of these things that in fact We do have that these are Euclidean domains now I have to caution you when it comes to a Euclidean domain to be a Euclidean domain You have to have that a norm is Euclidean Built into the definition of Euclidean norm excuse me Euclidean domain is not a specific norm It's possible that when you have a certain norm on a domain that norm might not be Euclidean But it could still be a Euclidean domain because some other norm makes it equivalent makes it a Euclidean domain So one has to be cautious because given any interval domain one could have countlessly be different non-equivalent norms, so When we talk about Euclidean domains, we don't necessarily have a specific norm in mind You could replace a Euclidean norm with another equivalent Euclidean norm Of course, there's going to be some canonical examples like we've seen right here But this actually makes it somewhat difficult to prove that a domain is Euclidean or not because if we struggle to find a Euclidean norm that doesn't mean it doesn't exist It just means that it might be hard to find and so to prove that an integral domain is not Euclidean often requires that we then Prove that that norm doesn't exist which can be a hard feat to do at times