DescriptionEuclidean domains are integral domains where we have a notion of division with remainder, and hence the Euclidean algorithm. As such, these rings play a fundamental role in number theory. In particular, it turns out that they all have unique factorisation into irreducible elements. The precise definition is: Definition. Let $R$ be an integral domain. A Euclidean function (or norm) on $R$ is a function $\phi:R\setminus\{0\}\rightarrow \mathbb{N}\cup\{0\}$such that
Examples of EDs include the ring of integers $\mathbb{Z}$ with $\phi$ given by $a\mapsto|a|$. (This is because we have division with remainder.) and the ring of polynomials $F[x]$ for any field $F$ with $\phi$ given by $f(x)\mapsto\deg f$ (division with remainder for polynomials over a field). It is also true that the Gaussian integers $\mathbb{Z}[i]$, as well as the ring $\mathbb{Z}[\sqrt{-2}]$ are Euclidean domains. There are also more exotic Euclidean domains and a host of variations and generalisations that we will study. For example, one can replace $\mathbb{N}\cup\{0\}$ by any well-ordered set (where every non-empty subset has a least element). It is not hard to prove that Euclidean domains are principal ideal domains (PIDs), and moreover, a more deeper theorem says that every PID is a unique factorisation domain (UFD). One of the main questions is to determine when the ring of integers in a number field is a Euclidean domain and, more specifically, whether it is Euclidean with respect to the absolute value of the norm coming from the number field (it is then called norm-Euclidean). We will define these notions as we go along. A ring of integers need to be a PID, but if it is, it makes sense to ask whether it is Euclidean. This is not always the case. We will see that the ring $\mathbb{Z}[(1+\sqrt{-19})/2]$ is not Euclidean, yet it is known to be a PID. On the other hand, we expect that as soon as the ring of integers has infinitely many units, then it is a PID iff it is Euclidean. This is because of the following theorem. Theorem (Weinberger, Hooley). Assume the Generalised Riemann Hypothesis. If a ring of integers of a number field has infinitely many units and is a PID, then it is Euclidean. It was only in 1994 that it was shown that a ring of integers can be Euclidean but not norm Euclidean: Clark showed that this holds for $\mathbb{Z}[(1+\sqrt{69})/2]$. Even more recently, in 2004, the second example was given $\mathbb{Z}[\sqrt{14}]$. It is known that there are only finitely many norm-Euclidean quadratic rings (essentially rings given by adjoining a square root, as above). In contrast, it is believed that there are infinitely many Euclidean quadratic rings, but that remains an open problem. Further directions for individual study include various generalisations of Euclidean domains (see Lemmermeyer article), studying the constructions of exotic Euclidean norms, like the one on $\mathbb{Z}[\sqrt{14}]$, as well as Lenstra's notion of Euclidean ideal classes (certain elements that sometimes exist in the ideal class group). PrerequisitesAlgebra IICo-requisitesNumber Theory III Resources
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email: Alexander Stasinski