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


Published:20040201
Printed: Feb 2004
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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:
\begin{center}
\emph{%
the ring of integers of $K$ is a Euclidean domain if and only if
it is a principal ideal domain.}
\end{center}
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 Motzkintype 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.