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ℤ[] is Euclidean

Published online by Cambridge University Press:  20 November 2018

Malcolm Harper
Malcolm Harper*
Affiliation:
Champlain College, St. Lambert, Québec, J4P 3P2 e-mail: malcolmharper@sympatico.ca
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Abstract

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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.

Keywords

11R0411R11
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 1 , 01 February 2004 , pp. 55 - 70
Copyright
Copyright © Canadian Mathematical Society 2004

References

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