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$\mathbb{Z}[\sqrt{14}]$ is Euclidean

Open Access article
 Printed: Feb 2004
  • Malcolm Harper
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We provide the first unconditional proof that the ring $\mathbb{Z} [\sqrt{14}]$ is a Euclidean domain. The proof is generalized to other real quadratic fields and to cyclotomic extensions of $\mathbb{Q}$. It is proved that if $K$ is a real quadratic field (modulo the existence of two special primes of $K$) or if $K$ is a cyclotomic extension of $\mathbb{Q}$ then: $$ the~ring~of~integers~of~K~is~a~Euclidean~domain~if~and~only~if~it~is~a~principal~ideal~domain. $$ The proof is a modification of the proof of a theorem of Clark and Murty giving a similar result when $K$ is a totally real extension of degree at least three. The main changes are a new Motzkin-type lemma and the addition of the large sieve to the argument. These changes allow application of a powerful theorem due to Bombieri, Friedlander and Iwaniec in order to obtain the result in the real quadratic case. The modification also allows the completion of the classification of cyclotomic extensions in terms of the Euclidean property.
MSC Classifications: 11R04, 11R11 show english descriptions Algebraic numbers; rings of algebraic integers
Quadratic extensions
11R04 - Algebraic numbers; rings of algebraic integers
11R11 - Quadratic extensions

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