A RMIN L EUTBECHER
Euclidean fields having a large Lenstra constant
Annales de l’institut Fourier, tome 35, no2 (1985), p. 83-106
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35, 2 (1985), 83-106.
EUCLIDEAN FIELDS HAVING A LARGE LENSTRA CONSTANT
par Armin LEUTBECHER
INTRODUCTION
In this note we present 143 new examples of Euclidean number fields K having a sufficiently large Lenstra constant. This constant, the maximal length of exceptional sequences in K, has been introduced by H. W.
Lenstra Jr in [4] which is the basic reference. Lenstra there derives bounds o^ which guarantee that an algebraic number field K with r real and s complex Archimedean primes is Euclidean provided M(K) > a^^/ldicl, where d^ is the discriminant of K. In [6] J. Martinet and the author gave several applications of Lenstra's method of exceptional sequences. This paper is a supplement to [6]. The search for fields having a large Lenstra constant again revealed for several signatures n, r fields K,^ whose discriminant is smaller in absolute value than those of all other examples known before (n = r + 25 is the degree).
This article ends with an updated table of the number of all known Euclidean fields counted according to their values of n and r + s respectively (table 5). It is made on the model of table 11 in [4] which had been updated in [5] and [6]. For n = 4, r + s = 2 the quartic subfield of the 13th cyclotomic field is counted, which was shown to be Euclidean by F. J. van der Linden using different methods [7]. There he also proves that this field and the 5th cyclotomic field are the only complex cyclic fields of degree 4 which are Euclidean with respect to the norm [7], Theorem 10.30.
22 of the new Euclidean fields are imprimitive. Acknowledgement is due to J. Martinet whose advice concerning the subfields I have been following at several places.
Key-words: Units and factorization - Class number, discriminant - Euclidean rings and generalizations.
1. THE GRAPH OF EXCEPTIONAL UNITS
1.1. General observations.
Let A b^ a commutative ring with 1 ^ 0 and R* its group of units.
Following [6] we call a sequence ©i, ©2, ..., CD^ of elements of R
« exceptional sequence » if o\ — o)y e R* for each pair of different indices i,j. This gives rise to the following construction. Two elements a, b e R shall be connected iff a — b e R*, thus R becomes a graph. The number M = M(R) of vertices of a maximal complete subgraph, i.e. the maximal length of exceptional sequences in case R = ZK is the ring of integers of an algebraic number field K, has been called the Lenstra constant of K.
We are so free to call it the Lenstra constant of the ring R in this general context. All translations and all multiplications by elements M e R* yield automorphisms of R as a graph. Therefore what M(R) is concerned we can restrict the consideration to maximal complete subgraphs which contain 0 and 1 as vertices.
Every ring morphism (p : R -^ R' having IR< = (pOp) also gives rise to a morphism of the graph of R into the graph of R\ Therefore
M(Ro) ^ M(R) ^ M(R/I)
for each subring Ro of R with the same unit element and for each ideal I ^ R. The elements u e R which are connected with 0 and 1 simultaneously will be called « exceptional units » following Nagell [9]. The set E(R) of all exceptional units of R is finite in case R = ZK is the ring of integers of an algebraic number field, a fact which to my knowledge was first proved by S. Lang [3].
The two transformations co »—^ 1 -- CD and CD »-+ 1/co are automorphisms of the subgraph E(R) of exceptional units. In general they generate a group of order six. Whenever E(R) is not empty we thus have a group G of order 6 isomorphic to the symmetric group 83 acting on E(R).
It is convenient to extend the subgraph {0} u R* by a further element oo, which is connected with each of these elements. On this graph R*
operates by multiplication and also G) H-» I/CD is an automorphism. So
^((0) == fl/G)
is an automorphism of R* u {0,oo} for each a e R * .
For each x e E(R) there is a Kleinian four group operating on the set of neighbours of x , given by its elements of order two:
^(o) = x/o), r,((o) = ^-j-5 ^(®) = ^ _ ^ •
1.2. Conditioned exceptional sequences.
We are considering number fields K = Q(x) where x is a zero of a monic irreducible / e Z[X] in some fixed algebraic closure Q of Q. For each ^eZ[X] the element g(x) is in the integral closure ZK of Z in K, however, g(x) € ZK is possible only for primitive polynomials g . In case g(x) is a unit the canonical ring morphism of Z[X] into ZK by substitution X ^ x extends uniquely to a ring morphism (py ofZ[X,l/g]
into ZK.
Starting with the ring of fractions Z[X, l/^] one gets for each complete subgraph S of Z[X,1/^] the complete subgraph (py(S) in ZK which has the same number of vertices as S has. This can be done with a fixed g for various /.
The product formula of the resultant R(/,g) in terms of the zeros x, of / and YJ of g shows that g(x) is a unit in ZK iff R(/,g) = ± 1.
Given a primitive geZ[X] and a complete subgraph S of Z[X,1/^] one has to look for monic polynomials F e Z[X] for which R(F,g) == ± 1.
Then one has R(/,g) == ± 1 for each monic irreducible factor / of R.
Similarly R(/,^) = ± 1 is equivalent to R(/,^i) = ± 1 for each primitive irreducible factor g^ of g . In case g^ is also monic and y is a zero of gi the condition R(/,^i) = ± 1 is the same as f(y) e Z^. This can be read as a system of m = deg g^ linear equations for the unknown coefficients of /, for which the right hand side depends on a parameter in the unit group Z$^. We shall speak of a condition of the first kind on /.
In aid of a short notation we introduce as in [6] the set U(/) of those integral y c Q , for which f(y) is a unit in ZQ^).
In case g^ is not monic, we call R(/,^i) = ± 1 a condition of the second king on /.
2. EXAMPLES IN THE NUMBER FIELD CASE
Starting with a suitable monic ^eZpC] and a complete subgraph S in Z[X,l/g] with M vertices we construct monic polynomials FeZpC] of given degree n and resultant R(F,^) = ± 1. Each irreducible factor / of F with a zero x e Q gives a field K = Q(x) and a complete subgraph (p/S) of ZK with likewise M elements. Let d^ denote the discriminant of ZK , r and 5 the number of real and complex Archimedean places of K respectively. If
M > o^^Kl
ZK is already shown to be a Euclidean ring. In case
M < O^^KI < L(K)
for the least ideal norm L(K) > 1 in ZK, we try to prove M(K) > 0^,/JdKl by enlarging <p^.(S).
In aid of our examples we have used different sets of conditions of the first kind on polynomials /eZ[X] which define the fields. For the purpose of formulating these conditions as well as for the description of certain subfields we need some concrete algebraic integers which are put together in table 1 (similar to and compatible with table 10 of [4]).
table 1
symbol minimal polynomial discriminant
9 X^X-l 5 a X^X-l -23 Y X^X-l -51 n X^X^X-l 49 0 X^X^X^X+l 1 1 7 p X^SX^X-l -275 6 X^X-l -285 c X^X^X-l -551 T X^X^X-l - 6 4 5
Besides i^ always denotes a primitive w — th root of unity.
To start with
S : 0, 1, x, x + 1, x2
is an exceptional sequence on condition 0, 1, — l,9eU(/).
T: 0,l,x,(x-l)/^,l/(l-x)
is an exceptional sequence on condition 0, 1, ^ € V(f). Both have been used in [4]. S u {00} consists of three orbits of <s^> and T u {00}
consists of two orbits of the subgroup <() of G whose generator ( is acting by
, , co- 1 r«o)=^-.
Several different enlargements of S have been studied in [6]. Thereby frequently use was made of the transformations s^ and q^+x' In this paper the larger part of examples is derived from enlargements of T especially by full orbits of < ( > . A smaller part is based on S. In aid of a space saving notation of exceptional sequences we need in both cases some abbreviations. Their relevance is restricted to the last column of the main table 2 and that of table 3. Let a = l/r(x), b == l/t(x-hl), c = l / t ( x2) , d = x 2 + x , e = x3 — x , beta = x4 — x3 — x2 + x -h 1 , eta = x3 -h x2 — 2x — 1 . We denote by a dash the s^ image : a' = s^(d) etc. With this notation the most useful sequences in [6] have been
A : S,a,a' B : S, b C: S,b\c,d.
In this paper nearly all examples in the S line are based on B* : B, c
which is an exceptional sequence on condition 0,1, - l,e,a,^2eU(/).
As for the T line in [6] the following sequence was introduced D : T, x - x2
which is exceptional on condition 0,1, t^, ^, a2 e U(/). The sequence D : T,x - x^x-x2), t'^x-x2)
is exceptional on one extra condition, namely P2— p e U ( / ) . Furthermore the following abbreviations are used
u = x/((x), v = x/((x—x2), w = x/C^x—x2),
^ = ^ ° ^ o q^x-x2), z = x/r(r).
Here qx° suo Ax viewed as a homographic map is given by the matrix
(^ ^)-
Thus it interchanges oo and x , 0 and t(x), 1 and t~l(x). The images of x — x2, ((x—x2), (^(x—x2) are y , t ~l( y ) and ^(^) respectively.
Therefore the sequence D', consisting of 0,1 and the t orbits of x and y is exceptional on the same conditions as D is. Similarly the homographic map (lx-x10 seo(lx-xl wt^ £ == (x—x2)/t(x—x2) is given by the matrix
(
x^x — x ^ x — l ^ -1 -x2-^ /It interchanges oo and x — x2, 0 and t(x—x2), 1 and ^ ~l( x — x2) , whereas the images of x,t(x),t~l(x) are \|t~l(y) = t ( l / y ) , l / y and l / t ( y ) = t ~l( l / y ) respectively. Therefore the sequence D" consisting of 0,1 and the t orbits of x — x2 and l / y again is exceptional on the same conditions as D is.
3. COMMENTS ON THE MAIN TABLE
3.1. Explanation.
Table 2 contains all Euclidean fields found since the publication of [6] with the exception of the quartic subfield of 0(^13) mentioned in the introduction. Also two polynomials are listed in degree n = 10 whose discriminants already appear in [6].
Table 2 consists of 5 parts. The first part contains fields K of degree n = 7. In column 1 one finds the number r of real places. Column 2 gives the discriminant d^ of K and its factorization. In the third column the coefficients OQ , a^, ..., a-j of an irreducible monic polynomial / are listed, a zero x of which defines K = Q(x). The next column contains a lower bound on M(K) which guarantees
M(K) > oc^^d
and the last column contains an exceptional sequence in ZK in terms of x, which shows the lower bound to be valid. — The other parts are explained similarly. Each of them contains only fields of a fixed signature n, r. An extra column headed by Kg gives a field generator 60 ofapropersubfield Ko of K of maximal degree if K is imprimitive, a blank otherwise. If K is imprimitive, the polynomial / defining K is in K()[X]. The field generator 60 is one of the symbols of table 1 or 9o = y / — 7 or in part 5 with quintic subfields Ko, 60 == 6^ is defined in the free space of the 3d
column.
We conclude this subsection by some of the bounds a^ used for the column headed by M >
n = 7 r = l , s = 3 and r = 3, s = 2: 9.848 10-3
r == 5, 5 = 1 : 7.793 10-3
n = 8 r = 0, s = 4 and r = 2, s = 3 : 3.955 10-3
r = 4 , 5 = 2 : 3.897 10-3 n= 9 r = l , s = 4 and r = 3, 5 = 3 : 1.563 10-3
n = 10 r = 0, 5 = 5 and r = 2, 5 = 4 : 6.097 10-4
3.2. Exceptional sequences for some extra fields in table 2.
For several fields K of table 2 the exceptional sequence has the shape
«So in S^rpbS2» where So is a subset of some conditioned exceptional sequence Si and 82 is a subset of ZK. This should be read «So replaced by 83 ». Furthermore, special sequences are used in the following cases:
Part 2 : n = 8, r = 2, d = - 5 371 171
O.l.x.xVjc2 - Ux+lV^x^x+lU^+x2^-!)/^2, (x4-^2)/^3-^-!),^4-^^2)/^3-^-!).
Part 3 : n = 8, r = 4, d = 15 297 613.
In [6] an exceptional sequence of length 14 was given, namely A,d,d\ 1 - &\5,(l-b'),^ -h 1, 0?+l)/x, s,(0?+l)/x).
This sequence can be enlarged by the following three elements x4 - 2x2 + 1, x3|(x3-x2-x+l), (x^-2x2-x+l)|(x2-x), a fact found by the student G. Niklasch on occasion of an advanced course at the Technische Universitat Miinchen in 1983.
Part 4 : n = 9, r = 1, d = 35 686 793
0, l,x,x + l,(x+l)/x2, 1/Oc2-!), - l|(x3-x2-x), (x3+x2-x-l)/JC2, (x^-2x2-x+l)|(x4-2x2),
^^2x2-x+l)/(x3-x2-x).
Part 5 : n = 10, r = 0, d = - 292693979.
This is an example to show how more complicated exceptional sequences were found. At first one has the exceptional sequence D, v. In the second step it was observed that the pair t ~1 (x — x 2), v can be replaced by the triplet w, t o q ^ o t '1^ ) , s^(t(x—x2)) and testing this new sequence again for possible enlargements it was found that the element x can be substituted by the two elements s^(x) t(Su(t(u))). Application of t~1 and rearrangement gives
0, 1 , X , ( X - 1 ) / X , X - X2, l/^-JC+l^JC/^+l), - l / ( x3- X2^ X - l ) ,
(x3- x2-l)/(x3-x2+x-l), (-x^^lx3-x2^x)|(x2-x+V), (x3-x2)|(x^-x3+2x2-2x+l).
By a similar procedure in the case of discriminant d = — 298 482 287 an exceptional sequence of length 11 was found : in D, v, t(y\ z,t~l(s^(t(x-x2))) the triplet t(x), v, t(y) can be replaced by
u,xls^(t(x-x2)), t(5^(x)), rOc/t-^Oc-jc2))).
3.3. Subfields.
Our construction of the fields K with prescribed exceptional sequences yields irreducible monic polynomials /eZ[X] defining K = Q(x) by a zero x of /. The discriminant of / may signal a possible proper subfield K o ^ Q of K.
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