Euclidean Ring

A Euclidean ring, also called a Euclidean domain, is an integral domain equipped with a function such that for every with there are a quotient and a remainder satisfying , with either or . The function is called a Euclidean function or Euclidean norm. It need not be multiplicative or equal to an absolute field norm (Lorscheid 2020, pp. 22-23). Successive divisions give the Euclidean algorithm. For the integers, the absolute value is a Euclidean function, and for the polynomial ring over a field , the polynomial degree of each nonzero polynomial can be used. Other examples include the Gaussian integers and .

Every Euclidean ring is a principal ideal domain and hence a principal ring. To see this, take a nonzero ideal and choose a nonzero minimizing . For every , division by gives with . Minimality forces , so (Lorscheid 2020, p. 24).

Weighted norms provide examples beyond the ordinary field norm. For , the exact range of weights making a particular weighted norm Euclidean is its Euclidean window (Cardon 2026).

Grobner (2026) announced that the ring of integers of every totally real field that is an Abelian number field is Euclidean iff its class number is 1. Euclideanity here does not require the field norm to be a Euclidean function. The argument explicitly takes major OpenAI analytic results as inputs pending verification, so the claimed general equivalence remains qualified by those unverified inputs.

For a ring of integers, using the absolute field norm as a Euclidean function defines a norm-Euclidean number field. Bagger et al. (2026) determined all 58 complex cubic fields with this stronger property.