Understanding the Motivation Behind Euclidean Domains

Question

To provide explicit questions:

1. What is the Euclidean Domain supposed to be a generalization of (with respect to its history)? In other words, was there a ring which motivated the initial construction of such an object? I've read from another post that Gaussian Integers were central to the creation of Euclidean Domains,... but other than this (very broad) fact, I've got nothing.

2. If you expect that the underlying history is a little bit beyond my reach at my current point in Abstract Algebra, all I need is a broad timeline of events (with specific years and names of mathematicians).

Motivation. In understanding underlying motivator in creating such a class of rings, my expectation is that then -- and only then -- should I be able to develop a better understanding as to why the "lax" conditions of Euclidean Domains (particularly regarding the Euclidean function) are sufficient.

I appreciate all the help I can get! Thank you.

Answer 1

By definition, generalizing the classical cases $\,\Bbb Z\,$ and $\,K[x],\,$ Euclidean domains are those domains enjoying division with "smaller" remainder. As in the classical cases, this immediately implies all ideals are principal, generated by any one of its smallest elements $\,a\neq 0$ $\,(a\,$ must divide every $\,b\in I,\,$ else the remainder $\,0\neq b\bmod a = \color{#0a0}{b-qa}\in I$ and is smaller than $\,a).\,$ Here "smaller" can be any well-ordering, but the classical cases use $\,\Bbb N\,$ (magnitude of integers and degree of polynomials).

The classical proofs in $\,\Bbb N, \Bbb Z, K[x]\,$ (for specific ideals, e.g. see these proofs of Bezout gcd identity and Euclid's Lemma and least denominator divides every denom, and $\,a^n\equiv 1\Rightarrow {\rm order}(a)\mid n,\,$ and $\,f(a)=0\Rightarrow {\rm min.poly}(a)\mid f$) $ $ were well known long before rings and their ideals were axiomatized, so they were presented in older language. Once ideals (or groups) were known, it was obvious how to reformulate these proofs at their natural level of generality, leading to the modern definition of a Euclidean domain.

In fact we can generalize such Euclidean descent to any PID. The Dedekind-Hasse criterion states that a domain $\,D\,$ is a PID $\!\iff\!$ given $\,0\neq a,b\in D,\,$ if $\,a\nmid b\,$ then some $D$-linear combination $\,\color{#c00}r\color{#0a0}{b\!-\!qa}\,$ is "smaller" than $\,a.\,$ The same descent proof as above shows such a domain is a PID. Conversely, since a PID is UFD, an adequate "smaller" measure is the number of prime factors (if $\,a\nmid b\:$ then their gcd $\rm\,c\,$ must have fewer prime factors, for if $\rm\:(a,b) = (c)\:$ then $\,c\:|\:a\:$ properly, else $\,a\:|\:c\:|\:b\:$ contra hypothesis). See also Hurwitz's division algorithm.

In summary, the concept of a Euclidean domains arose by abstracting into ring ideal language the common arithmetical structure in many classical "descent by remainder" proofs.