J
OURNAL DE
T
HÉORIE DES
N
OMBRES DE
B
ORDEAUX
R
OLAND
Q
UÊME
A computer algorithm for finding new euclidean
number fields
Journal de Théorie des Nombres de Bordeaux, tome
10, n
o1 (1998),
p. 33-48
<http://www.numdam.org/item?id=JTNB_1998__10_1_33_0>
© Université Bordeaux 1, 1998, tous droits réservés.
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
A
computer
algorithm
for
finding
neweuclidean
number fields
par ROLAND
QUÊME
RESUME. Cet article donne la
description
d’unalgorithme
infor-matique
fournissant une condition suffisante pourqu’un
corps de nombres soit euclidien pour la norme, ouplus
brièvement eucli-dien. Dans le recensement des corps euclidiens et des méthodes de recherche deceux-ci,
FranzLemmermeyer
amentionné,
[3]
p405,
que 743 corps de nombres euclidiens étaient connus(mars
1994), (le
premier
d’entre eux,Q
découvert parEuclide,
troissiecles avant
J.-C.!).
Durant lespremiers
mois de1997,
nous avons trouvéplus
de 1200 nouveaux corps de nombres euclidiensde
degre
4,
5 et 6. Ces résultats ont été obtenusgrace
à unalgo-rithme
informatique
mettant en oeuvre lesproprietes classiques
des
réseaux, plongements
de 1’anneau des entiers d’un corps denombres K de
degre n
dansRn ,
ainsi que la structure du groupe des unités de K. La fin de cet article est unegeneralisation a
larecherche des anneaux euclidiens de S-entiers des corps de
nom-bres K et à l’étude du minimum
inhomogene
de la forme Norme. Les resultats obtenus sont coherents avec ceux de labibliographie.
ABSTRACT. This article describes a computeralgorithm
which exhibits a sufficient condition for a number field to be euclidean for the norm. In the survey[3]
p405,
FranzLemmermeyer pointed
out that 743 number fields where known
(march 1994)
to beeu-clidean
(the
first one,Q,
discoveredby Euclid,
three centuriesB.C.!).
In the first months of1997,
we found more than 1200 neweuclidean number fields of
degree
4,
5 and 6 with a computeral-gorithm involving
classical latticeproperties
of theembedding
of thedegree n
field K into Rn and the structure of the unit group of K. This articles ends with ageneralization
of the method for the determination ofrings
ofS-integers
of number fields euclidean for the norm and for thestudy
of theinhomogeneous
minimum of the norm form. Our results are in accordance with known results.1. SOME GENERALITIES
1.1. Definitions on number fields. In this
section,
we define theprin-cipal
mathematicalobjects
used in theproofs
and in theC++
program.Let II8 be the field of real numbers. Let C be the field of
complex
numbers. Let K be a number field ofdegree
n definedby
a monicpolynomial
P(X)
withintegral
coefficients,
thus K = withP(x)
= 0.Let us define the
signature
(rl , r2 )
of the number fieldK,
where ri is the number of real roots ofP(X) =
0 in C and2r2
is the number ofcomplex
roots of
P(X) = 0 in C.
Letaci, i
=1, ... , n
be the roots ofP(X) = 0 in
C. In the
sequel,
these roots will be ordered in this way : xi, i =1, ... ,
r1
are the real roots of
P(X)
= 0 and r2 are thecomplex
roots ofP (X ) -
0,
where Xrl+2i = =1, ... ,
r2 . Thediscriminant of the number field K is noted D.
We define the
embedding
K #
x~’’2 , by
the relationand, concurrently,
we note also the nconjugates
of xby
the relationsThen,
thebijective
map x C2-~
is definedby
Finally,
the third map T is defined as thecomposition
map of theprevious
ones:Note
that,
if the field K istotally real,
then Tl is theidentity
map and a = T. In thisarticle,
we note A thering
ofintegers
of the number fieldK =
~(x) .
The is a basis of K. We define anintegral
the relations :
1.2. Classical
geometric properties
of number fields. Thispara-graph
states some basicgeometric
results and notations used in thesequel.
We use thenotation p
=(pl, ... , p~,)
for thepoints
of Themap T defined in the relation
(5)
allows us togive
ageometric description
of thering
ofintegers
A. The set TA =r(A) =
{T(a) la
EA}
is a latticeA basis of the lattice is the set of n
points
The fundamental domain F of the lattice TA is the set of
points
p E I~n whichverify
the relations p =ribi
- +ribi
-f- +Tnbn,
whereri
verify
theinequalities
1 i n.1.3. Geometric definition of the norm form. Let c be an
algebraic
number,
c E K. The norm of c is definedby
the formulaFrom the
previous
relation,
we deduceimmediately
that :The norm
form,
orshortly,
the norm of apoint
p of RI is the formgiven
by
the formula :From the two relations
(9)
and(10),
we deducethat,
for c EK,
we have theidentity connecting
thealgebraic
andgeometric point
of view :1.4. Definition of fundamental units. The set of units of
A,
notedA*,
is a group of Z-rank r =rl + r2 - 1.
Any
unit - E A* can be written :where 770
generates
the group of roots ofunity
in K and f/1, ... , fIr is a setof r fundamental units.
1.5. Definition of euclidean number fields. The number field K is said
to be euclidean
if,
for all c EK,
there exists q E A such thatq)
I
1 oN(T(c - q))
1 «N(T(c) - T(q))
1. For K to beeuclidean,
a sufficientgeometric
condition isthat,
for all p ER~,
there exists q E A such asWe shall use that sufficient
geometric
condition in thesequel :
~ we say that a
point
p E Rn is euclidean if there exists q E A such thatN~P ‘ T (q) ) 1.
~ we say that a subset G of W is euclidean if for all
p E G,
there existsq E A such that
N( p - T (q) )
1.~ Note
that,
if the fundamental domain F of K iseuclidean,
then IEgn iseuclidean and the number field K is also euclidean.
2. PRINCIPLES OF THE ALGORITHM
The fundamental domain F is covered
by
a set of small cubes of R~ and thealgorithm
checks over theeuclideanity
of each of themby
differentangles
of attack. If the fundamental domain F is coveredby
a set of cubes alleuclidean,
the number field K is euclidean. Note that thealgorithm
exhibits sufficient condition for a number field to beeuclidean,
but not atall a necessary and sufhcient condition.
Stage
1 of thealgorithm
searches for euclidean cubesonly by geometric
considerations.Stage
2 of thealgorithm
searches for euclidean
cubes,
among those left indeterminate instage 1,
using
thegeometric properties
of theembedding
T of the units of thering
ofintegers
A in3. STAGE 1
3.1. Some definitions.
Here,
are somespecific
definitions and notations of thispart
of thealgorithm :
9 Let p E RI and a E I~. Let
C(p, a)
denote the cube ofR7,
of centerp, of
edge length
a, of which the npairs
of faces areperpendicular
to9 Let Let
d(pi,p2)
denote the distance between PI and p2given
classically by
the formula :. Let 0 be the
origin
of Let F be a fundamental domain ofT(A)
of which theorigin
0 is a vertex. F is aparallelotope.
Letbl, ... , bn
be the basis of the lattice TA definedby
(7).
Let s be thesymmetry
center of theparallelotope
F. LetP; (F)
be the face of F definedby
the set of npoints
{O,
b1,... ,
bi- 1, bi+ 1, ... ,
Let be the second face of Fparallel
toPi(F).
LetPi (H)
be theplane
ofRn,
parallel
toPi(F),
at a distance J =x f
ofPi(F)
and not in the same side ofPi(F)
than s. LetPn+i(H)
be theplane
ofparallel
to at a distance J =
~
x 9
of and not in the same side of than s. Let H be theparallelotope
of of which faces are the 2nplanes
i =1,...,
n.3.2. Some theoretic
propositions.
This section contains some theoreticpropositions
used in theprogramming
ofstage
1 of thealgorithm.
Proposition
1. Thefundamental
domain F issymmetric
withrespect
tothe
point
T(bs)
=If
thepoint
p is
euclidean(respectivedy
not
euclidean),
then thepoint
T(2bs) -
p is
euclidean(respectively
noteu-clidean).
Proof.
The fundamental domain F is aparallelotope
built on theorigin
0 and the basis{bl, ... , bn}
of the lattice TA.Classically,
thisparallelotope
issymmetric
withrespect
to thepoint
r(bs)
=T( 61+’2+bn ).
If p iseuclidean,
then there exists q E A such that
N(p -
T(q))
1 ~N((T(2bs) -
p)
-TA ~
T (2bS ) -
p is an euclideanpoint symmetric
to the euclideanpoint
p.The
proof
is similar for non euclideanpoints.
0Therefore,
in thealgorithm, by
symmetry,
we shall limit the determi-nation ofeuclideanity
of the fundamental domain F to half apart
of this domain and thus reduce the CPU timeby
2.Proposition
2. LetC(0, a)
be a cubeof
henceof
center 0 andof edge
length
a. Let p be apoint
Then,
for
allpoints u
EC(0, a),
we havewhere
This
inequality
is the bestpossible
in the sensethat,
for
a vertexof
C(0, a),
Proof.
Let u EC(0, a).
We have theinequalities :
which leads to relation
(15).
Theequality
iseffectively
obtainedwhen,
for i =1,...
n, we have[ = g
andsign(ui)
= therefore the resultis the best
possible.
0Proposition
3. LetLa
={ila, ... , ina
I
il
EZ, ... , in
EZ}
be the latticedefined from
a,edge
length of
the cubeC(0, a).
Let I be the(finite)
setof
points
of
La
flH,
where theParallelotoPe
H isdefined in
the section(3.1).
Then,
the set Jof
cubesI
Pi EI}
covers thefundamental
domain F.Proof.
If0,
then there exists apoint y
E F such thata #
because the euclidean distance 6 of anypoint
of the cubeC(0, a)
to the center of this cube is smaller thanThen,
from the definitions of theparallelotopes
F andH,
we deducethat Pi
E H.There-fore,
the set J of all the cubes whose centerbelongs
toLa
n H cover thefundamental domain F. D
3.3.
Stage
1 of thealgorithm.
In thissection,
we describe the firststage
of thealgorithm.
Let K be a numberfield,
with the fundamental domain F of the lattice 7 A. To prove that K is an euclidean numberfield,
it is sufficient to prove that all the cubes of the set J(defined above)
are euclidean. Let a cubeC(p, a)
E J. To prove that the cubeC(p, a)
iseuclidean,
it issufficient, using
theproposition 2,
that thereexists q
E Averifying
the relationDetermination of euclidean cubes
C(p, a).
Recall thatC(p, a)
denotes the cube of ofcenter p
ELa
n H and ofedge length
a. The candidates for are taken from the set of latticepoints
withd (T (q), o)
m¡n,
verifying
relation(18),
with m --~ +oo. Thecomputer
algorithm
takes thegreatest
values mpossible
which arereasonably compatible
with the CPUspeed
of thecomputer.
A trick of the
algorithm
is,
when we fail tofind q
with euclidean cubeC((p, a),
consists intrying again locally
for thiscube,
with apartition
ofC(p, a)
in smaller cubes ofedge length
2 ,
orThen,
at the end of thestage
1 of thealgorithm,
the alternative is :~ All the cubes of J are
euclidean,
and the number field K is euclidean.~ For m cubes c
J, j =
1, ... , m~,
thealgorithm
stage
1 does not allows us to conclude ifthey
are euclidean :then,
we say that these m cubes are indeterminate.4. STAGE 2 OF THE ALGORITHM
In this
section,
we describe thepart
of thealgorithm involving
the units of the number fieldK,
toreduce, and,
ifpossible
toannihilate,
the number of indeterminate cubes. The method is similar to the oneexplained by
Stefania Cavallar and FranzLemmermeyer
in[1]
for cubic fields.4.1. Definitions. Recall that the map a, Tl and T are
respectively
definedby
the relations(2), (4)
and(5).
We fix some notations and definitions for thispart
of thealgorithm:
~ Let
Cj =
C(pj,
a),
j
=l, ... ,
m be theremaining
indeterminatecubes at the end of
stage
1 of thealgorithm.
~ Let E E A* be a unit of the
ring
A.~ Let d =
{dl, ... , dn}
be apoint
of From the relation(4),
we havewhere i is the
complex
number,
i2
+ 1 = 0.9 Let c E K. If
[
1 with q EA,
then]
1. Inthe same way, if
q)~ >
1 for all q EA,
thenq)1 ~
1 for all q E A. Themultiplication by
a unlit 6 of A* lefts invariant theset of non euclidean
points
ofr(~).
This remark leads us to define a mapr (K) 4
T(.K)
by
=r(bê).
Let d E I~n . The map -y, can be extended to the map
R’ 4
by
the relationFrom
= 1,
it can be seen that thetransformation -Y, keeps
the volumes of· We note, in the
sequel,
thecompact
of R~ definedby :
Note that the volume in RI of is
equal
to the volume in ofCj
because the map 7tkeeps
the volumes.Call size of a
compact
C CW,
the real s definedby s
=sup(d(x, y))
for all x, y E C. The size of the
compact
r j (e)
isgenerally
muchlarger
than the size of the cubeCj
and even of the fundamental domain F of because i =1, ... ,
rl +r2)
is,
generally,
much4.2. Some theoretic results. The cubes
Cj, j
=1, ... , rrz
are leftin-determinate
by
stage
1.Generally,
some of them are in facteuclidean,
sometimes all. Thestage
2 of thealgorithm
aims toreduce,
or toannihi-late,
the number of cubes left indeterminate instage
1.Stage
2 rests on the three nextpropositions using
thegeometric
properties
of the fundamental units.The
compact
rl(e),
1 =1, ... , m
is said euclidean if all itspoints
areeuclidean. is euclidean if and
only
if the cubeC,
is euclidean because 7, lefts invariants the units of A* . The subsetT () }
of]Rn contains all the non euclideanpoints
of In the same way, the subsetcontains all the non euclidean
points
ofTherefore,
if a cubeCj
has a non euclideanpoint,
there exists at least one1,
1 L m and one a E A such thatOJ
n(r~(~) - 7-(c~)) ~
0.
On the otherhand,
ifci n
(rl(e) -
T (a) ) ~ =
0,
we conclude thatCj
iseuclidean,
which leads to the nextproposition:
Proposition
4.If
we have the relationthen the cube
Cj
is euclidean.The two next
propositions
aim to find cubesCj
verifying
the relation(22)
to eliminate them qua euclidean cubes.Proposition
5. Let m be the numberof
indeterminate cubesCj
=C(pj,
a)
at the end
of
stage
1of
thealgorithm.
Let the cubeCj
and thecompact
ri (e) -
A necessary condition on a E Afor
Cj
fl(fl(e) - -r (a)) 0 0
is thatr( a) =
~ .. ,art +2r2)
E Rnverzf y
theProof.
To have the relation(rl(e) -
0,
it is necessary that there exists u EC(0, a), v
EC(0,
a)
and a E A whichverify
the relation. For i
=1, ... ,
r1: we deduceFrom this relation and from the
inequalities
andivil
]
2 ,
we can deduce the relation(23).
o For i
= 1, ... ,
r2: From the relation(28),
we haveThen,
we obtain from the relation(20)
and theprevious
relation(30)
the relation inC,
From this relation and of the
inequalities
R,
f,
and we can deduce the relations
(24),
(25), (26)
and(27).
0 4.3. Remark. If the
edge
length
a ofC(pj,
a)
is smallcompared
to the fundamental domainF,
there exists at most one a E Averifying
the propo-sition(5).
4.4. Remark. The
proposition
(5)
shows that theintegers a
of Averi-fying
(23),(24),(25),(26)
and(27)
if any, are contained in aparallelotope
T whose faces areperpendicular
to axesRl’)
of Rn and of which the re-lations(23), (24), (25), (26)
and(27)
gives
the vertex coordinate on In thecomputer
algorithm,
the determination of theintegers
a E A such thatbelongs
to thisparallelotope
is obtained in this way: Wecom-pute,
for k =l, ... ,
2’ the coordinates i =l, ... ,
n of the vertexof T on the
integral
basis Y ={T ( y~ ), ... , T ( yn ) ~ ,
andthen,
for eachi,
1 i n,
the minimum rrai = k =1, ... ,
2’~ andmaxi-mum
Mi
=max(tk(i)),
k =l, ... ,
2n.Let,
for a EA,
theexpression
ofT(a)
on theintegral
basis Y:al, a2, ... , an E Z. The
possible
values ai are boundedby
the relationswhere the number of candidate a is finite
because ai
E Z.Let the cube
C~
and thecompact
forj, l given. Then,
let a E Averifying
the necessary condition ofproposition
(5).
The nextproposition
gives
acriterium,
easily implemented
on acomputer,
to determine ifcj n (rl(e) -
T(a)) -
0.
Infact,
we define a discrete set ofpoints
C such that n
OJ = 0 =>
ri (e)
ncj =
0.
Proposition
6. Let the cubeOJ
and thecompact
given.
Let a E Averif ying
the relations(~3), (,~,~~, (~5), (~6)
of
theproposition
(5).
Let pi ER,
i= 1, ... ,
n be elementsdefined by
the relations:Let
Ml
be thefinite
discrete setof points
~Cof
defined by
the relationsProof.
We examinesuccessively
the case of real andcomplex embeddings:
. At
first,
consider the realembeddings i
=1, ... , rl :
the minimaldistance Ji
on theith
coordinate of RI between twopoints
of isgiven
by
the relation6i
=pilai(e)l.
Therefore,
from therelation
(33),
.
Then,
consider thecomplex embeddings
(r1
+ 2i -1, r1
+2i), i
=1, ... ,
r2: Note that the square ofedge length
a of +2i-1) X +2i)~ C ~
I~2,
whose the vertex coordinate are(0,0), (a, 0), (0, a), (a,
a)
is transformedby
themultiplication
_ , . _ .. , _ . ,
on a square of
edge length
alurl+2i-1(e)1 ]
# the minimal distance~rl+2i-1
between twopoints
of on the coordinatejR(rl +2i-l)
x]R2
isgiven by
the relationðrl+2i-1 =
IUrl+2i-1(e)1
x whichleads,
from the definition of Prl+2i-1, as for the realembeddings
toTherefore,
the minimal distance~?
i =1, ... ,
rl between two
points
ofa real
embedding
on or the minimal distance6r~ +2j-1 , I = 1 , ... , r2
between twopoints
of acomplex embedding
onR(Il +2i-1) X
Rl+2i) N
~g2
isalways 1.
Then,
from the definition of and from the relations(35)
and(36),
we deduce that if(ri (e) -
T(a))
n C(pj,
a) # 0,
there is at least onepoint
of(Ml(ê) - T(a))
E #(Ml(~) -
0,
which leads to the result :
If
(Ml(E) -
T(a))
has nopoints
in the cubeC(pj, a),
thenr(a))
nC (pj,
a)
=0.
04.5.
Description
ofstage
2 of thealgorithm.
At the end of thestage
1 of thealgorithm,
the alternative is : all the cubesC(p, a)
covering
F are euclidean and the number field K iseuclidean,
or there are m cubesa),
, j =1,
... , m which are left indeterminate:stage
1 did not allow us to conclude that these cubes are euclidean.Stage
2 of thealgorithm
aimsto find some euclidean cubes among them. Let the indeterminate cubes
~C(pj,
a)
( j
=1, ... ,
?7~}.
Thealgorithm
examines each ofthem, for j
=1,...,
m.Suppose j
and I aregiven.
If there are not any a E A whichverify
the relations(23), (24),
(25),(26)
and(27)
ofproposition
(5),
then,
for all a EA,
(rl(e) - T(a))
a) =
0
=* thepair
(j, 1)
can be eliminated. If not, we can then find a such that the relations(23), (24), (25), (26)
and(27)
are verified. Noticethat,
if we suppose that theedge
length
a of the cubes C issufficiently
small,
a isunique.
Then,
weapply
theproposition
(6) :
if we have(Ml(~) -
T(a))
n C(pj,
a) = 0,
we can eliminate thepair
( j, 1).
If,
for j
given,
we can eliminate all thepairs
( j, 1),
l =1,...,
m,then the cube is euclidean from the
proposition
(4).
We have made no
assumption
on the unit E, thus thisalgorithm
can be used with all units - E A*. It is CPU timeconsuming
if the unitis of
large
size,
moreprecisely
if =1, ... , n
is alarge
realnumber,
for instancegreater
than146.
Therefore,
we limit thealgorithm
to a set of fundamental units of small size and their inverseI
, i =1, ... ,
T1+~2 2013 1}.
If there are no cubesC (pj,
a)
left indeterminateby
stage
2 of thealgorithm,
the number field K is euclidean.5. COMPUTER RESULTS
This section
gives
the results obtained in the first semester of 1997 for the number fields ofdegree
n, 4 n 6 with theC++
program described in this article. The number fieldK,
generated
by
a root x of the polyno-mialP(X),
of discriminantD,
ofintegral
basis{yl, ... , yn}
expressed
in function of the basis{1,
x, ... , ofK,
with the set of fundamental units{~1, ... ,
expressed
in function of theintegral
basis are found in the files of the Servermegrez . math . u-bordeaux . f r/numberf ields/
of the Bordeaux
University.
When,
in thefollowing results,
the discriminants are notedD1,
D2,
... ,that means that in the
given degree
andsignature,
there are several fields K with the same discriminant D.Among
them,
65 are new,compared
with the listgiven by
Franz Lem-mermeyer in[3].
14400,
14515,14591,
14703, 14843, 14896,
14975 are leftindeterminate 1
in thecomputer
test we made2.
The other 681 fields are norm-euclidean.Among
them,
664 are new,compared
with the listgiven
in[3].
5.3. Number fields n =
4,
r1 =4,
r2 = 0.Among
the 283 fields with0
37532,
the field with D = 21025 is notprincipal
thereforenot
euclidean,
the 26 fields with discriminant D =13725, 15125, 16400,
17725,
18432,
22000, 23525, 24336, 26225, 27725,
28400,
30125, 30400,
32225, 32625, 33625, 33725, 34816, 35152, 35225,
37525, 37952, 38000,
38225,
38725,
38864. are indeterminate. The 256 other fields are euclidean.Among
them,
239 are new,compared
with[3].
5.4. Number fields n =
5,ri = 3, r2 =
1. The two fields with D =-18463, -24671
are indeterminate for a trivial reason! the Bairstow method of resolution ofP(x)=0
for thepolynomial
P(X)
has failed in thetest).
All the other 92 fields with discriminant
D,
0 37532 are norm-euclidean.Among
them,
at least 82 are new,compared
with[3].
5.5. Number fields n =
5,ri
=1, r2
= 2. The field with D = 16129 isindeterminate for the same reason. the Bairstow method of resolution of
P(x)=0
for thepolynomial
P(X)
fails in thetest)
All the other 146 fields with 0D
17232 are euclidean.Among
them,
at least 132 are new,compared
with[3].
5.6. Number fields n =
5,n
=5, r2
= 0. The 25 fields with 0D
161121 are euclidean.
Among
them,
22 are new,compared
with[3].
5.7. Number fields n =
6,ri
=0, r2
= 3. The 5 fields with9747 f
I D
11691 are euclidean.
5.8. Number fields n =
6,ri
=2, r2
= 2. The 11 fields with28037
35557 are euclidean.
5.9.
Synthesis.
There are at least 1204 newfields, compared
with[3],
where the number of euclidean fields known in 1994 was743,
withdegree
n, 1n12.
1 The meaning of this word is given previously in the article
2with the initial parameters chosen, especially the value of the edge a of the cubes a) covering the fundamental domain F : it is possible, that for different initial parameters , some
6. SOME GENERALIZATIONS
We
give
here twogeneralizations
of thealgorithm:
. the
study
of euclideanrings
ofS-integers
As
of number fieldsK,
with. the
study
of theinhomogeneous
minimum of the norm formN( p-T (q) )
andq E A.
6.1.
Rings
ofS’-integers
As
of number fields K. Let v~l, ... , vpt be a set of t non archimedean valuations of K. LetAs
be thering
ofS-integers
of the number field Kcorresponding
to this set ofvaluations,
therefore such that for allvp 0
and for all a EAs ,
we have 0. O’Meara’s theorem assertsthat,
for allrings
As
definedby
a valuation set8 =
fvpl
it isalways possible
to find a finite set of valuations 8’ ={v, ... ,
with g Cg’
such that thering
As,
is euclidean for the norm, see for instance O.T. O’Meara in[5].
The Minkowski Bound of a number field .K is the constant Bgiven by
the formulaThe
quantitative
form of O’Meara’s theorem assertsthat,
if there exists m > B with sl, ... , sm E Ski for 1~~
k’,
(s~ - Ski)
EAS,
group of units of
As,
then thering
As
is euclidean for the norm, see for instance H.W. Lenstra in[4].
The nextproposition
aims toenlarge
thealgorithm
we haveexplained
to therings
As
of K.Proposition
7. LetC(pj,
a),
j =1, ... ,
t be a setof
cubescovering
thefundamental
domain Fof
the lattice TA. Let M E N. Let 8 be the setof
valuations
corresponding
to all theprimes
idealsof
A above aprime
pof
Nverif ying
1p
M. LetAs
be thering of
Kcorresponding
to the setof
valuations S.Then,
for
As
to beeuclidean,
it issufficient
that,
for
allj,
1 j
t,
there exists onevalue kj
EN,
1 k~
M such that the cubekja)
is euclidean.Proof.
Let kj
E Nverify
thehypothesis.
We deducethat,
Vu EC(O,
kja)
there exists q E A such that
N(kjpj
+ u - T(q))
1Vul
EC(O, a),
there exists q E A such that +ul ) - T (q) )
1.But,
if1 1~~
M,
it results from the definition of thering
As,
that kj
E(As)*.
Therefore,
forall c E
K,
it ispossible
to find - E(As)*
and q
E A CAs
verifying
the relationN(T(ce) -
T(q))
1,
which is a sufhcientcondition,
forAs
to benorm-euclidean. 0
6.2. Remark. Note that this
proposition
iseffectively
moregeneral
than the sufficient condition that we considered for thering
ofintegers
A to be euclidean : it ispossible
that,
forj, pj
given,
thealgorithm
fails to find theof the cube
C(kjpj, kja)
for one value M.Then,
it ispossible
that,
for someM,
thealgorithm
conclude thatAs
is euclideanthough
it leaves Aindeterminate,
which is confirmedby
the tests of thecomputer
program.6.3.
Example
of results. Let us consider theexample
of thecomplex
cubic fieldsK, n =
3, r, =
1, r2 =
1.Among
the fields withD, 0
! D ~
492,
the fields withIDI
=199,
283, 307,
327, 331,
335,
339, 351,
364, 367,
436,
439,
459,
491 are noteuclidean,
see for instance[3].
Thecomputer
program result showsthat,
for all these discriminantsD, 0
[
492,
then the fractionring
AS(2)
isnorm-euclidean,
where the set of non archimedean valuations 8corresponds
to the set of allprimes
above theprime
ideal 2Z of Z. The Minkowkibound,
for this set of number fieldsis bounded
by
MB
fi, 276.
Therefore,
with thequantitative
version of O’MearaTheorem,
thering
AS’(6)
isnorm-euclidean,
where the set of non archimedean valuationsg’ (6)
corresponds
to the set of allprimes
above theprime
ideals2Z,
3Z and 5Z ofZ, which, clearly,
is lessprecise
than theprevious
result.6.4.
Inhomogeneous
minimum of the norm form. Tostudy
thein-homogeneous
minimum of the norm form with thecomputer
algorithm,
we haveonly
toreplace
in the program the conditionN( p - T (q) )
1by
N( p - T (q) )
M,
where M > 0 isgiven.
To dothat,
we haveonly
tochange
one instruction of the C++program!
6.5.
Example
of results. Let .F~ be a number field ofdegree
n, ofsigna-ture
(~1,7*2)
and of discriminant D. H.Davenport
has shown in[2],
that for all number fieldsI~,
we have theinequality
where ~y is a
positive
constantdepending
on nonly.
Consider,
as anexample,
the fields K with :Some of them are left indeterminate for
euclideanity
in the section ofcom-puter
results. We have verifiedthat,
for all of them we have the relationwhere B is the Minkowski Bound of the field K. For these
fields,
this result is better than the result ofDavenport:
in that case, our result is on~
instead ofD 2n-2T2 _
Perhaps,
it would bepossible,
to formulate ageneralization
of the Min-kowskiconjecture
fortotally
real fields(N(p - T(q))
2"~B/D) :
for all number fields K with(rl, r2) 0 (0,1),
we should haveAcknowledgements
It is ourpleasure
to express ourgratitude
to Prof.Jacques
Martinet and Henri Cohen forhelpful
advice on this work and toFranz
Lemmermeyer
for manysuggestions during
all theperiod
of concep-tion and oftesting
of theC++
algorithm.
Theconsistency
of the results of Stefania Cavallar and FranzLemmermeyer
in[1]
for cubic fields and of FranzLemmermeyer
for thequartic
fields with our results waslargely
used for the validation of thealgorithm
and of the C++ program.REFERENCES
[1] S. Cavallar and F. Lemmermeyer, The euclidean algorithm in cubic number fields, draft (August 1996).
[2] H. Davenport, Linear forms associated with an algebraic number field, Quarterly J. Math.,
(2) 3, (1952), pp. 32-41.
[3] F. Lemmermeyer, The euclidean algorithm in algebraic number fields, Expo. Mat., no 13 (1995), pp. 385-416.
[4] H.W. Lenstra, Euclidean number fields of large degree, Invent. Math., n0 38, (1977), pp. 237-254.
[5] O.T. O’Meara, On the finite generation of linear groups over Hasse domains, J. Reine Angew. Math. n0 217 (1965), pp. 79-108.
[6] P. Samuel, Theorie algébrique des nombres, Hermann, (1967). Roland QUBME
13, avenue du chateau d’eau 31490 Brax
France