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MATHEMATICS OF COMPUTATION
VOLUME 46. NUMBER 174
APRIL 1986, PAGES 659-666
A Lower Bound for the Class Number
of Certain Cubic Number Fields
By Günter Lettl
Abstract. Let ATbe a cyclic number field with generating polynomial
i
a— 3 ^
x3 —Y-x1
û+ 3
-=~-x-
i
and conductor m. We will derive a lower bound for the class number of these fields and list all
such fields with prime conductor m = (a1 + 21)/A or m = (1 + 21b2)/A and small class
number.
1. Introduction. Let hm denote the class number of the cyclotomic field Q(Çm), and
A*, the class number of its maximal real subfield Q(cos(27r/w)). It is a well-known
conjecture of Vandiver that p 1 hp holds for all primes p e P. This is a customary
assumption for proving the second case of Fermat's Last Theorem (for more details
see Washington [16]). Since hp grows slowly (hp = 1 for p < 163 with the use of
the Generalized Riemann Hypothesis (GRH), van der Linden [10]), for no p with
hp > 1 the exact value of hp is known without using GRH. Masley suggested that
perhaps hp < p always holds, but a counterexample was found in [3], [12]. The class
number of each real subfield of Q(f ) divides hp, and in this way one can find
primes with hp > 1. Using the quadratic subfield, Ankeny, Chowla and Hasse [1]
showed that hp > 1 if p belongs to certain quadratic sequences in N, and S.-D.
Lang [9] and Takeuchi [15] found more such sequences. Similar results were
obtained for h4p by Yokoi [17]. Using the cubic subfield of Q(f ), which has been
thoroughly investigated (e.g., [2], [5], [8]), the theorem of the present paper yields the
following results:
If a is an odd integer, a > 23, andp = {a2 + 27)/4, a prime, then hp > 1.
Ifb is an odd integer, b > 1, andp = (1 + 27A2)/4, aprime, then hp > 1.
A conjecture about primes in quadratic sequences (Hardy and Wright [7, 1.2.8])
implies that there exist infinitely many primes p of each of these two forms, because
one can write
a2+
27
/--*\2
+ 3 ^z—
+9
and
I±™!_3(^lJL12 + ^±zl
i
Received November 6, 1984; revised July 15, 1985.
1980 Mathematics Subject Classification.Primary 12A30, 12A50.
©1986 American Mathematical
Society
0025-5718/86 $1.00 + $.25 per page
659
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660
GÜNTER LETTL
2. Class Number Bounds and Main Results. Let K be a cyclic cubic number field
with conductor m and class number A. It is well known that m is the product of
distinct primes, which are congruent to 1 mod (6), and of 9, if 3 ramifies in K. The
class number A is congruent to 1 mod (3), if m is a prime or m = 9, and A is
divisible by 3 otherwise. Set f(s) = L(s, x) ■L(s, x) for s£C, where x and x are
the non trivial cubic Dirichlet characters modulo (m) belonging to K. Since the
discriminant of K equals m2, the analytic class number formula yields
(i)
y>
h = ^iM
4R
'
where R is the regulator of K. Moser [11] showed that for prime conductors,
A < m/3 holds, so cubic fields will never lead to a contradiction to Vandiver's
conjecture. Our aim is to establish a lower bound for the class number of a special
family of cubic fields and to list all fields of some special types with prime conductor
and small class number. From a result of Stark [14] one can deduce /(l) > c/\ogm,
where c is effectively computable, but this bound is not suited for our purposes.
From the results of the next section we will obtain:
(2)
If K is a cyclic cubic number field with conductor m > 105, then
/(l) > 0.023 • w"
The harder problem is to find an upper bound for the regulator, which is only
achieved for the following family of cyclic cubic fields. The polynomial
(3)
fa(X) = X3-^X2-^X-l,
AENodd,
is irreducible over Q, has discriminant D(fa) = ((a2 + 27)/4)2, and if e is a zero of
fa, the other zeros are e' = -l/(e + 1) and e" = -(e + l)/e. Therefore, fa is a
generating polynomial of a cyclic cubic field K with conductor m, and we define
k e N by JD(fa) = (a2 + 27)/4 = km.
We call the field K of type A, if k = 1, and of type B, if k = 27 and a = 21b with
b e N odd, b # 1 (in this case we have m = (1 + 27b2)/4). It is well known that
fields of type A or B have relatively large class numbers (see, for example, the tables
of Gras [5]). Shanks [13] states that for cubic fields of type A with prime conductor
"a rough mean value for A is given by A « 12w/35(log m)2 ".
Lemma 1. Let K be a cyclic cubic field with generating polynomial fa, conductor m
and regulator R. Then,
(4)
4R<(\\o%D(fa))1
= (\o%(km))2.
Proof of Lemma 1. Since the zeros e, e', e" of fa are units of K, we can estimate the
regulator of K by R < Reg((e,e'}) =:R\
Choosing
a - 3 + 4\[km ■cos(l/3
e =-ikm
if R' # 0 (see Lemma 4.15 in [16]).
• arctan(v/2T/a))
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,-
THE CLASS NUMBER OF CERTAIN CUBIC NUMBER FIELDS
661
with the principal value of arctan, we obtain
R' = det(log|a|
\log|e'|
logls'l
log|e"
= (log|e + 1|)2 - log|e + l|log|e|
+(log|e|)2.
Series expansions yield
R' = \(\oêkm)2 - ^áM
+ _3_ + 0f tog_*m
2km
4km
\ (km)1
and elementary calculus explicitly gives (4).
With (2) and Lemma 1 we immediately obtain from (1):
Let K be a cyclic cubic field with conductor m > 105 and with generating polynomial
fa- Then,
A > 0.023
(5)
Theorem,
m 0.946
(log km)2
(a) Let K be a cyclic cubic field of type A with prime conductor m. Then
A < 16 holds onlyfor the followingvalues of m:
A
m
1
4
1,13,19, 37, 79, 97,139
163, 349,607,709, 937
313,877,1129,1567,1987,2557
1063
7
13
(b) Let K be a cyclic cubic field of type B with prime conductor m. Then A < 43
holds only for the following values of m:
m
1
4
7
13
28
31
37
61, 331
547,1951
2437,3571
9241
4219, 25117
23497
8269
Proof of the Theorem. From (5) we obtain A > 14 for fields of type A with
m ^ 169339, and A > 37.2 for fields of type B with m > 106. It is well known (see,
e.g., Gras [4]) that primes q = -1 mod (3) divide the class number of a cyclic cubic
field only with an even exponent. The table of class numbers of Shanks [13], and
Table 1 below, complete the proof of the theorem.
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GÜNTER LETTL
662
Table 1
Class numbers of cyclic cubic fields of type B
with prime conductor m < 106
1 + 2762
3
7
9
17
19
23
25
33
35
37
39
45
59
61
91
95
105
115
117
123
129
131
137
147
159
61
331
547
1951
2437
3571
4219
7351
8269
9241
10267
13669
23497
25117
55897
60919
74419
89269
92401
102121
112327
115837
126691
145861
170647
1 + 27¿2
1
1
4 = 22
4 = 22
7
7
173
185
189
191
193
199
28 = 22
49 = 72
37
13
49 = 72
109
31
28 = 22 1
133 = 7 • 19
193
205
221
227
231
235
243
259
261
297
299
301
688 = 24 43
211
532 22 7 ■19
303
307
347
361
367
604 = 22 151
148 = 22 37
97
652 = 22 163
628 = 22 157
305
341
371
373
383
202021
231019
241117
246247
251431
267307
283669
329677
347821
360187
372769
398581
452797
459817
595411
603457
611557
619711
627919
784897
812761
879667
909151
929077
939121
990151
316
79
343
1216
175
73
26 19
247 = 13-19
196 = 22 • 72
541
316 =
331
1732 =
553 =
1075 =
769
2257 =
2299 =
739
889 =
22 • 79
22 • 433
7-79
52 • 43
37-61
ll2 - 19
7-127
1156 22 ■172
1552 2" ■97
688 ' 24-43
769
688
24-43
787
1588 • 22 ■397
661
532 22 • 7 ■19
The class numbers of Table 1 were calculated with a " Sirius 1 Personal Computer",
using the analytic class number formula (1). We also used that for fields of type B
the roots of fa are already fundamental units, and therefore R = R' can be
calculated with the explicit formula for e, given in the proof of Lemma 1. In the
following way it can be proved that e is a fundamental unit:
Let AT be a field of type B with generating polynomial fa, a = 21b and
m = (1 + 27¿>2)/4. Hasse [8] investigated the arithmetic of cyclic cubic fields, using
the Gauss sums of the corresponding Dirichlet characters. With Hasse's notation,
every integer a G K can be written as a = [x, y] with x G Z, ye. Z[p], where
p2 + p + 1 = 0, and x = y mod (1 - p). If a is a unit of K, N(a) = 1 implies
x3 = 27 mod (m) and |jc| ^ 2^'myy (Satz 8, [8]). For the roots of f21b we have
e = [(27A - 3)/2, 3/V3 ] and its conjugates. Since Godwin's conjecture about fundamental units holds for cyclic cubic fields with m > 9 (see Gras [6]), we have to show:
There exists no unit a = [x, y] G K, a ¥= +1, with myy = \ tr(a — a')"
< j tr(e - e')2 = 27w, where tr denotes the trace from K to Q.
Suppose
the contrary.
Then
x3 = 27mod
(m)
and
\x\ < 2^2Jm
imply x g
{3,(27Z>
- 3)/2, -(276 + 3)/2} for b > 7. Considering0 ■ x =ymod (1 - p) and
yy < 27 yields only a few possibilities for y g Z[p], and one can check that for each
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663
THE CLASS NUMBER OF CERTAIN CUBIC NUMBER FIELDS
of these y, N(a) = 1 has no solution a ¥= 1. For small values of b, one can consult
the table in [5].
In the same way, but with much less computation, one can prove that for k = 1
(type A) and k = 3 the roots of fa are also fundamental units. In these cases one has
e = [(a- 3)/2, +1] with (a - 3)/2 = ±1 mod (3), and e = [(9b - 3)/2,/73],
respectively.
3. A Lower Bound for L(l, x) ■L(l, x). Let m be the conductor of a cyclic cubic
field K, x and x the nontrivial cubic Dirichlet characters modulo m associated with
K, and f(s) = L(s, x)L(s, x)- To find a lower bound for /(l), we first need an
upper bound for |/(*)| in a disk in C containing 1. Consider C = C(/t, p) =
[s e C\\s — p\ < p) with 1 < n and ¡jl- 1 < p < ¡i, and set a0 = ¡u,- p. Let s =
a + it e C For a > 0 we have the representation
L(s,x)= m£^+s.
n= l
f^ldx
"
Jm
withS(*,x)=
X
E x(«)
l<n<x
(see [16, p. 211]). The inequality of Pólya-Vinogradov [16, Lemma 11.8] states that
|S(*>X)I < i/w • logm. For s g C(ja, p), the function \s\/a = l/cos(args) attains
its maximum ¡i/ /ju2 - p2 if 5 is the point of contact of a tangent of C through 0.
Combining these results, we obtain for every s G C{\i,p):
(m
\L(s, x) \ < 1 + /
1
/—
/"°°
"";r ^ + \s\\m ■logm 1
<-ml~°a
+
^
logw
-
1
dx
• w05"°°.
Since logx/ v^x is monotone decreasing for x > e2, we conclude that for m > m0 >
(6)
|/(5)|
<q-m2-20"
holds for all i g C(p, p), with
_I
/
1
log^o \2j
M
\1 - °o ' {^
p2
im
Lemma 2. If K is a cyclic cubic number field with conductor m, thenf(l) > c6 ■m~Cl,
with c6, c-j > 0 as given in the course of the proof. Furthermore, c7 can be made
arbitrarily small.
Proof of Lemma 2. The proof follows mainly Washington [16, pp. 212-214]. Let
f(s) be the Riemann zeta function and ÇK(s) = ¡¡(s)f(s) the zeta function of the
cyclic cubic field K with conductor m. If s = a + it g C, we have
U*) = l+
00
a
E -7
fora>l,
n= 2
with a„ > 0, and an > 1 if n is a cube. Developing f^ in a power series around
jtt > 1 gives
SM-
Lbj(p-s)j,
j-0
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664
GÜNTER LETTL
with
1
(7) b0 = £M>
°°
a
í(3/i) > 1 and 6, = 77 E (log«)7 • -* > 0 for y > 1.
The integral representation of f(s) for o > 0 yields
roo
|f(*)l< s-
1
+ il/"00
s f-
1
A
j
aw
=
5-
1
and
|f(*)l< s<
+1-1-Ê^r-r1 (-!•» du
1
5-
n-1
"
+ Mi,
2 1 + ä1
1
Let C = C(jii, p), with ju - 1 < p < ¡ti, be the disk with center ¡i and radius p, and
denote its boundary by dC. Using (6), we get for all s G 3C:
1
SM s-/(i)1
1/(1)1< ¿2 m 2-2o
(8)
i-i
<lf(*)l•!/(*)!
+
with
H +1
lí- Il
c2 = c, • max
se3C
+ \s \ ■min
(MR)})-
Since ÇK(s) - f(l)/(s - 1) is holomorphic in the whole complex plane, (8) holds for
all s g C(ju, p). Computing the coefficients of
/(I)
(/i-l)
;-o\
7+7h(M-^
with a Cauchy integral gives
H^'-Ä)
/(I)
*;-
(m-1)
7+1
ds
£2
. w2-2o0
P'
(,-M)y+1
For 0 < a < 1, the integral representation of f(s), and f(a) = \L(a, x)|2, show that
^k(°)
< 0. So for any a with a0 < a < 1, and any y G R + with 1 < v, we have
/(I)
«-1
"-1
h!
-c2 • w22-2o
2o° • V^
E If
_
(ri)'
7TT -(^-«)J
_ « w
/(l)
f(l)(p-a
a —1
1 — a \ ju — 1
C • «r 2-2.JM-«
a - an
[-1-1
where E fy-ÎM~~a)y > c3 > 1.
y-o
From this inequality, we obtain
1 \ V
/(l)
> c3(l - a)
M- 1 1
ju — a
— Ct ■ m
2-2o0
.
P2(l - «)
(/i - a)(a
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ÍM-1
- o0) \
P
THE CLASS NUMBER OF CERTAIN CUBIC NUMBER FIELDS
665
Choosing v = c4 ■logm + c5, with
2 - 2a0
C4
p
log
¡u — a
and
log(, - Ik« - q0)+ lo^y~[
- ]°^y^i
log- P
H- a
gives /(l)
> c6 • m"'7, with
C< CÁl
aV-W
C2 (,-a)(a-a0)
[
p
and
c7 = (2-2a0)-log^/log^.
Since c7 -» 0 for a -* 1, the proof of Lemma 2 is completed.
Numerical computations show that for m0 = 105 good results can be obtained by
choosing ft = 10, p = 9.9 and a = 0.975. With these values we obtain c2 = 10.8685
and v ~ 315.
Using (7), and an > 1 for n a cube, we obtain the following estimations for c3:
["1-1
300 -,
oo
E Vm - a)J > U/0 + E -7 E -*((m- «)iog«)J
7= 0
/=i ■'■ n = 2 "
W0
-,
300 ,
>í(3/0 + E7ÍE-T((M-«)3-log^)y
A= 2 *
> t , V
j=\J-
1 í,.«.-»
((M - «)3 • log*)"1
*-»*^\
301!
302
|
' 302-U-a)3-logfc]'
where 7V0< e302/3(/x-<»)\yith the special values for p., p and a, and 7V0= 100, we
obtain
[cl-l
100
E ä,-(m- «)' > E *~2925 - 10"40 > 1-2175 = c3.
7=0
k=l
These values yield c6 > 0.023 and, c1 < 0.054, and thus (2) is proved.
Institut für Mathematik
Karl-Franzens-Universität
Halbärthgasse 1
A-8010 Graz
Österreich, Austria
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"Arithmétique des corps abéliens du troisième degré," Ann. Sei. Ecole Norm. Sup.
(3), v. 63, 1946, pp. 109-160.
3. G. Cornell
& L. C. Washington,
"Class numbers of cyclotomic fields," J. Number Theory, v. 21,
1985, pp. 260-274.
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666
GÜNTER LETTL
4. M.-N.
Gras,
Sur le Nombre de Classes du Sous-Corps
Cubique de Q''''
(p m 1 (3)), Thèse,
Grenoble, 1971.
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des extensions cubiques cycliques de Q," J. Reine Angew. Math., v. 277,1975, pp. 89-116.
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und biquadratischen Zahlkörpern," Abh. Deutsch. Akad. Wiss. Berlin Math.-Nat. KL, 1948, Nr. 2, 95
pp.
9. S.-D. Lang, "Note on the class-number of the maximal real subfield of a cyclotomic field", J.
ReineAngew. Math., v. 290,1977,pp. 70-72.
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v. 39,1982, pp. 693-707.
U.C.
MOSER, Sur le Nombre de Classes d'un Corps K Réel Cyclique de Conducteur Premier, deg K > 4,
Séminaire Théorie des Nombres, Grenoble, 1980,17 pp.
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with an application to real cyclotomic fields", Math. Comp., v. 41, 1983, pp. 303-305.
13. D. Shanks, "The simplestcubic fields," Math. Comp.,v. 28,1974, pp. 1137-1152.
14. H. M. Stark, "Some effective cases of the Brauer-Siegel Theorem," Invent. Math., v. 23, 1974, pp.
135-152.
15. H. Takeuchi,
"On the class-number of the maximal real subfield of a cyclotomic field," Canad. J.
Math.,\. 33,1981,pp.55-58.
16. L. C. Washington, Introduction to CyclotomicFields, GTM 83, Springer, New York, 1982.
17. H. Yokoi, "On the diophantine equation x2 - py2 = ±Aq and the class number of real subfields
of a cyclotomicfield," NagoyaMath. J., v. 91, 1983,pp. 151-161.
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