maximal systems of independent units in arbitrary number fields. The tables in .... steps two of these numbers are associated with a nontrivial unit e¿ satisfying. (2.2) .... W_1 < |^jc|q(o¿)I < |o4J')|C1(n_1)Ar
MATHEMATICS OF COMPUTATION VOLUME 52, NUMBER 185 JANUARY 1989, PAGES 149-159
Computation of Independent Units in
Number Fields by Dirichlet's Method* By Johannes
Buchmann
and Attila
Pethö
Abstract. Using the basis reduction algorithm of A. K. Lenstra, H. W. Lenstra, Jr. and L. Lovász [8] and an idea of Buchmann [4], we describe a method for computing maximal systems of independent units in arbitrary number fields. The tables in the supplements section display such systems for the fields Q( \ÍD) where 6 < n < 11.
1. Introduction. Let K be an algebraic number field of degree n > 2 over Q, let R be an order in K and let E be the group of units of R. The structure
of E was described in 1846 by Dirichlet [6]. He proved that if K has s real and 2< nonreal conjugate fields, then E is the direct product of the finite group of the roots of unity in E and r = s + t —1 infinite cyclic groups. In the sequel we assume r>l. Dirichlet's proof was based on his diophantine approximation theorem: Let cvi,...,an G R, n > 2; then there exist for any Q G R, Q > 1, integers Xi,...,x„ which are not all zero such that
\xí\ 2, |7[J)| > \l[Jli\
for j € {1,... ,8 + t},j ¿i,k>
2.
Obviously, these numbers have to be pairwise distinct, and after a finite number of steps two of these numbers are associated with a nontrivial unit e¿ satisfying
(2.2)
|tf} | < 1 and \e¡3)| > 1 for j ^ i.
It is well known that every subsystem of cardinality s + t —1 in {ei,... maximal system of independent units in R (cf. [9]). The sequence ("ik)keN is constructed
as follows: To initialize the sequence, we
set
(2.3)
,£s+t} is a
7i = 1.
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COMPUTATION OF INDEPENDENT
UNITS IN NUMBER FIELDS
151
Now suppose that we know 7*. Then we define
(2.4)
Rk = ~R,
Nk = \NKlQ(lk)\,
Ik
and using techniques of diophantine approximation, the module Rk satisfying
(2.5)
\ßil)\\ßi3)\>l
we compute a number ßk in
íorJG{l,...,s
+ t},j¿i,
and \NK\Q(ßk)\/4 • Dn>2 ■/c-n("-2>/2,
and thus (3.3) and (3.20) yield |/3W|2 < D ■«TÍ"-2) < C2/c-(n-2>JV-2/",
(3.24)
|x,| < 2(n2+1)/(4("-2)),c
for 3 < I < n.
An upper bound for xi and x2 follows from
|xi Re 4
+ x2 Re a2 \ = Re/?«-¿x,Re4i) 1=3 n
\xi Im4
+ x2Im4 I — Im/^-^Tx/Inm}0 ¡=3
Applying Cramer's rule, we get in view of (3.5), (3.23) and (3.24),
(3.25)
|x¡| < C^kN-^D'1
4. Computational
Aspects
< C3k
tor I = 1,2.
of the Algorithm.
D
Let i G {l,...,s
again a fixed conjugate direction. Before we give a detailed algorithm, we give some preparatory explanations.
description
+ í} be of the
Assume that we know for a k G N the number Nk = \NK\Q(ik)\ and an LLLreduced basis ai(k),... ,an(k) of the module Rk = R/^jkIn order to compute the number /?* satisfying (2.5), we have to proceed as follows: - choose k, - set 6 according to (3.14) or (3.18), - set U according to (3.13) or (3.17), - apply the LLL-algorithm to the columns of U resulting in U = U ■T, with T = (U,j)i 0.**
Output: The unit £¿.
1. Initialization. a¡(l)
C
OO^HO
CN -^
0*s O O "^
°^LAl0
Ocsl
V7V-
3-,">r? III
*Fd«f
HHHCD
I^HQH
O
1*
»T1
rt O r-l (
xT1
vv,
'Si
3 LA
■o
a.
P5
□
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