The Euclidean algorithm for number fields and primitive roots

Abstract

Let $K$ be a number field with unit rank at least four, containing a subfield $M$ such that $K/M$ is Galois of degree at least four. We show that the ring of integers of $K$ is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann Hypothesis for Dedekind zeta functions. We prove this unconditionally.

Locations

• Proceedings of the American Mathematical Society - View - PDF
• CiteSeer X (The Pennsylvania State University) - View - PDF