The Euclidean algorithm for number fields and primitive roots

Murty, M.; Petersen, Kathleen

doi:10.1090/S0002-9939-2012-11327-9

The Euclidean algorithm for number fields and primitive roots
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by M. R. Murty and Kathleen L. Petersen PDF

Proc. Amer. Math. Soc. 141 (2013), 181-190 Request permission

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.

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