Euclidean Ring

A Euclidean ring is a ring without zero divisors in which an integer norm and an associated division algorithm (i.e., a Euclidean algorithm) can be defined. For signed integers, the usual norm is the absolute value and the division algorithm gives the ordinary quotient and remainder. For polynomials, the norm is the degree.

Important examples of Euclidean rings (besides ) are the Gaussian integers and , the ring of polynomials with complex coefficients. All Euclidean rings are also principal rings.

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.