Re: Euclidean ring

From: Gerry Myerson (gerry_at_maths.mq.edi.ai.i2u4email)
Date: 03/08/05 

Date: Tue, 08 Mar 2005 13:57:19 +1100

In article <1110229345.675886.169740@o13g2000cwo.googlegroups.com>,
 "Alon Amit" <alon.amit@gmail.com> wrote:

> On that topic, I vaguely recall a claim that the Euclidean status of
> Z[sqrt(14)] was an open problem a few years ago. Anyone knows if that's
> still the case?

It appears that Malcolm Harper has shown that Z[sqrt(14)] is Euclidean.
See his paper in Canad. J. Math. 56 (2004), no. 1, 55--70.

http://journals.cms.math.ca/CJM/v56n1/index.en.html

Here's the abstract:

 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. 

-- 
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)

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