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# Euclidean Rings of Algebraic Integers

Published:2004-02-01
Printed: Feb 2004
• Malcolm Harper
• M. Ram Murty
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## Abstract

Let $K$ be a finite Galois extension of the field of rational numbers with unit rank greater than~3. We prove 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 now prove this unconditionally.
 MSC Classifications: 11R04 - Algebraic numbers; rings of algebraic integers 11R27 - Units and factorization 11R32 - Galois theory 11R42 - Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 11N36 - Applications of sieve methods

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