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Here we give the results that we have obtained, and that, as far
as we know, are new.\\[4mm]
$\bullet$ Legend :\\[2mm]
- $K$ number field of degree $n$,\\
- $M(K)$ Euclidean minimum of $K$ which is equal to its inhomogeneous minimum [C2],\\
- when $n=2$, $K={\Bbb Q}(\sqrt{m})$,\\
- when $n=3$, E indicates that $K$ is norm-Euclidean.\\
- $T$ number of critical rational points in a fundamental domain,\\[4mm]
$\bullet$ Table ``deg2'' : $n=2$.\\[2mm]
For $n=2$ we complete the tables that can be found in [L] and that give the
Euclidean minima of $\q(\sqrt{m})$ for $2\leq m \leq 102$,
where $m$ is a squarefree integer.
We have computed $M(K)$ for $K=\q(\sqrt{m})$, $m$
squarefree and $103\leq m\leq 400$, with 28 exceptions, for $m=$139,
151, 163, 166, 191, 199, 211, 214, 239, 241, 249, 262, 271, 283,
307, 311, 313, 319, 331, 334, 337, 358, 367, 379, 382, 391, 393
and 394. These cases have not been treated because of the size
of the fundamental unit ($\vert \varepsilon\vert>10^7$). Of
course, this can be done in multi-precision.\\[2mm]
In each line we give $m$ ($K=\q(\sqrt{m})$), the number $T$ of
critical rational points in a fundamental domain and $M(K)$.\\[4mm]
$\bullet$ Table ``deg3a'' and ``deg3b'' : $n=3$.\\[2mm]
For $n=3$, we complete the results obtained in [CL].\\[2mm]
In Table ``deg3a'', we give the Euclidean minima of cubic fields
that were not computed in [CL]. If we did not compute the Euclidean minimum,
essentially by lack of time, we
give an interval in which it is. The notations are the same as for Table ``deg2''.\\[2mm]
Table ``deg3b'' is devoted to number fields of discriminant less than 11000
that were left indeterminate in [CL] (in fact, they are all norm-Euclidean),
and to all the number fields of discriminant more than 11000 and
less than 15000, not studied in [CL]. We just give the Euclidean minimum for number fields which
are not norm-Euclidean. For others the letter E indicates that
they are norm-Euclidean.\\[4mm]
$\bullet$ Table ``deg4'' : $n=4$\\[2mm]
Same notations as in Tables ``deg2'' and ``deg3a''.
For $n=4$ we have computed Euclidean minima of the 286 number fields of discriminant
less than 40000.
The Euclidean nature of a large number of them has already been
found in [Q] but Qu\^eme had left some fields
indeterminate. In fact, some of these last
ones are not norm-Euclidean, although they have class number one (for $D_K=$ 18432, 34816 and 35152).\\[4mm]
$\bullet$ Table ``deg5'' : $n=5$\\[2mm]
Same notations as before.\\
Here there were just 25 number fields known to be norm-Euclidean [Q].\\
We have computed the Euclidean minima of the 156 number fields of
discriminant less than 511000. With one exception, the field $K$ of
discriminant 390625 which has class number one but verifies $M(K)=7/5$, they are all norm-Euclidean.\\[4mm]
$\bullet$ Tables ``deg6'', ``deg7'' and ``deg8'' : $n=6,\,7$, and $8$.\\[2mm]
Same notations as before.\\
For $n\geq 6$ as far as we know, very little was known on the fields of
degree greater than 5. We have treated the 156 first number fields for $n=6$
and the 132 first number fields for $n=7$. They are all norm-Euclidean.
Until now, we have not used the algorithm in a systematic
way for degree 8. Nevertheless we have computed the Euclidean
minimum of the 18 first fields given in Kl\"uners's tables [K],
which are all norm-Euclidean. Recall that we had already treated the maximal
real subfield of the cyclotomic field $\q\left(\zeta_{32}\right)$,
$K=\q\left(\sqrt{2+\sqrt{2+\sqrt{2}}}\right)$ [C1], whose
Euclidean minimum is $1/2$. We have also found with
our algorithm,
that, if $K=\q \left(\sqrt{2+\sqrt{2+\sqrt{3}}}\right)$, we have
$M(K)=\minho=1/2$.\\[6mm]
$\bullet$ References :\\[2mm]
- [C1] {\sc J-P. Cerri}, De l'euclidianit\'e de $\Bbb
Q\left(\sqrt{2+\sqrt{2+\sqrt{2}}}\right)$ et
$\Bbb Q\left(\sqrt{2+\sqrt{2}}\right)$ pour la norme,
{\em J. Th. Nombres Bordeaux} {\bf 12} (2000), 103--126.\\[2mm]
- [C2] {\sc J-P. Cerri}, Euclidean and inhomogeneous
spectra of number fields with unit rank greater than $1$,
(to appear in {\em Journal f\"ur die Reine und Angewandte Mathematik}).\\[2mm]
- [CL] {\sc S. Cavallar and F. Lemmermeyer}, The Euclidean algorithm
in cubic number fields, in Gy\H{o}ry, Peth\H{o}, Sos eds.,
Proceedings Number Theory Eger 1996, de Gruyter, 1998, 123--146.\\[2mm]
- [K] {\sc J. Kl\"uners}, Tables available at \url
{http://www.mathematik.uni-kassel.de/~klueners}.\\[2mm]
- [L] {\sc F. Lemmermeyer}, The Euclidean algorithm in algebraic
number fields, {\em Expositiones Mathematicae} {\bf 13} (1995), 385--416.\\[2mm]
- [Q] {\sc R. Qu\^eme}, A computer algorithm for finding new
Euclidean number fields, {\em J. Th. Nombres
Bordeaux} {\bf 10} (1998), 33--48.
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