Acta Mathematica et Informatica Universitatis Ostraviensis
Gérard Boutteaux; Stéphane Louboutin
The class number one problem for the non-normal sextic CM-fields. Part 2
Acta Mathematica et Informatica Universitatis Ostraviensis, Vol. 10 (2002), No. 1, 3--23
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Acta M a t h e m a t i c a et Informatica Universitatis Ostraviensis 10 (2002) 3-23
The class number one problem for the non-normal
sextic CM-fields. Part 2.
Gerard Boutteaux and Stephane Louboutin
Abstract. We delineate the determination of all the non-normal sextic CMfields with class number one and whose maximal totally real subfields are nonnormal cubic field. There are 367 non-isomorphic such fields. Since we had
already proved elsewhere t h a t there are 19 non-isomorphic non-normal sextic
CM-fields with class number one and whose maximal totally real subfields are
cyclic cubic fields, we are now in a position to conclude that there are 386
non-isomorphic non-normal sextic CM-fields which have class number one.
1. Introduction
Lately, great progress have been made towards the determination of all the normal
CM-fields with class number one. Due to the work of S.-H. Kwon, Y. Lefeuvre,
F. Lemmermeyer, S. Louboutin, R. Okazaki, Y.-H. Park and Y.-S. Yang, all the
normal CM-fields of degrees less than or equal to 48 with class number one are
known (with only partial solutions in the special cases of normal fields of degree 32
and 48). In contrast, up to now the determination of all the non-normal CM-fields
with class number one and of a given degree has only been solved for quartic fields
(see [LO]). The present piece of work is an abridged version of half the work to be
completed in [Bou] (the PhD thesis of the first author under the supervision of the
second auhtor): the determination of all the non-normal sextic CM-fields with class
number one, regardless whether their maximal totally real subfield is a non-normal
totally real cubic field (the situation dealt with in the present paper) or a real cyclic
cubic field (the situation dealt with in [BL]).
Let K be a CM-field, i.e. K is a totally imaginary number field, hence of even
degree 2n > 2, and K is a quadratic extension of its maximal totally real subfield
k, hence k is of degree n. We let hK, QK £ {1, 2} and WK denote its relative class
number, its Hasse unit index and the number of complex roots of unity contained
Received: January 9, 2002.
1991 Mathematics
Subject Classification: 11R16, 11R29, 11R42.
Key words and phrases: sextic field, CM-field, class number, zeta function.
Gérard Boutteaux and Stéphai
in K, respectively. Note t h a t WK = U>A where A is the maximal abelian subfield of
K. We have (see [Wa, Chapter 4]):
(I)
h-
-
QKWK
f~<~-es'=i(CK)
K
where dK and dk
respectively.
~
(2ir)« V dk Hes.-i(Cfc) '
denote the absolute values of the discriminants of K a n d k,
2. On CM-fields of odd class numbers
Proposition 1. Let K be a CM-field of degree 2n, n > 1 odd, and let TK/k
the finite part of the conductor of the quadratic extension
K/k.
denote
(1) At least one prime ideal of k is ramified in the quadratic extension
K/k.
Therefore, dK > 3d| and the narrow class number h^ of k divides the class
number hK of K. Consequently, if hK is odd then h~\ is odd, h^ = hk and
every totally positive unit of k is the square of some unit of k.
(2) Assume that hK is odd. Then, exactly one prime ideal Q of k is ramified
in the quadratic extension K/k and QK = 1- Finally, if Q is above an odd
rational prime q, then TK/k = Q, Q = 3 (mod 4) and the inertia degree f
of Q is odd.
Proof.
(1) Let TK/k denote the finite part of the conductor of the quadratic extension
K/k and let \ b e t h e quadratic, character associated with this quadratic
extension K/k. According to class field theory, there exists some primitive
quadratic character \o on the multiplicative group (Ak/TK/k)*
(where Ak
denotes the ring of algebraic integers of k) such t h a t for any a € Ak we
have x((&)) = v(a)xo(oi) where v(a) € { — 1} denotes t h e sign of the norm
NK/Q(&)
of a. In particular, if K/k is unramified at all t h e finite places
of k then for any algebraic unit e of k we must have v(e) = + 1 . Taking
n
e = —1 for which v(e) = ( — l ) , we obtain that n must be even.
(2) Let t denote t h e number of primes ideals of k which are ramified in t h e
quadratic extension K/k. According t o the first point of this Proposition,
we have t > 1. Assume t h a t hK is odd. Since 2l~l divides hK (see [Lou3,
Proposition 6]), we have t — 1. Moreover, since 2l divides hK if QK = 2
(see [Lou3, Proposition 6]), we have QK = 1. Now, if g is odd, then K/k is
tamely ramified, hence TK/k — Q- Since the multiplicative group (Ak/Q)*
is cyclic of order qf - 1 and since v(-l) = ( - 1 ) " = - 1 (by t h e first point),
we have 1 = x ( ( - 1 ) ) = .v(-l)Xrj(-l)
= ~ x o ( - l ) = -(-l){qf
~1)/2, which
yields qs = 3 (mod 4). Hence, q = 3 (mod 4) and / odd. •
Lemma 2 . Lev K/k = k(y/a)/k
a is an algebraic integer of k.
(4a)=l2TK/k-
be a quadratic extension of number fields, where
There exists some integral ideal 1 of k such that
P r o o f . Let Ak denote t h e ring of algebraic integers of k and VK/k denote t h e different of the quadratic extension K/k. Since VK/k = gcd{(2 % /a); a e Ak and K —
Non-normal sextic CM-fields
k(y/a)} (see [Lan, Chapter III, Prop. 8]), for any a e Ak such t h a t K = k(s/a)
there exists an integral ideal XK of K such t h a t (2y/a) = X2KVK/k- Taking relative
norms, we do get (4a) = X2NK/k(VK/k)
= X?TK/k where X = NK/k(XK).
•
Proposition 3 . Let K be a CM-field of degree 2n, n > 1 odd, let TK/k
the finite part of the conductor of the quadratic extension K/k, assume
class number hK of K is odd, let Q denote the unique prime ideal Q of k
ramified in the quadratic extension K/k (see Proposition I) and let q >
the rational prime below Q.
(1) 7 / g > 2 thenTK/k
= Q.
(2) If q = 2 and \ / - T £ K, then there exists r > 1 odd such that TK/k
(3) If [q = 2, s/^l £ K and ^ q $ KJ or %f [q > 2 and y^q & K],
neither totally ramified in k/Q nor inert in k/Q.
denote
that the
which is
2 denote
= Qr.
then q is
P r o o f . Let a be any totally positive algebraic element of k such t h a t K = k(y/—a).
There exists some integral ideal X of k such that (4a) = X2TK/k,
by Lemma 2. Set
h; := h^ = hk, which is odd by Proposition 1.
(1) K/k is tamely ramified.
(2) If TK/k
= Qr with r even, then (4a) = J2 with J := XQr/2.
Since
h is odd and since both J2 and Jh are principal in the narrow sense,
J is principal in the narrow sense and there exists some totally positive
unit c of k such t h a t 4 a = e/32. Since e is the square of some unit of k
(see Point 1 of Proposition 1), we obtain 4 a = 7 2 for some j G k and
K = k(y/^a)
= fc(v/-T), a contradiction.
(3) Assume t h a t (q) = Qn is totally ramified in k/Q or that (q) = Q is inert
in k/Q. Since TK/k = Qr for some odd r (by the previous two points)
and since n is odd, we obtain (4ga) = J2 for some integral ideal J of
k. As in the previous point, we obtain 4qa = j 2 for some 7 € fc and
K = k(\/^a)
= k(y/=:q), a contradiction. •
3. Bounds on residues
Lemma 4 . Let K be a totally imaginary number field of degree 2n > 2. Set Xn =
n ( 7 + log(27r)) — 1, where 7 = 0.577 • • • denotes Euler's constant. Then, ~ < 0 < 1
and CK(0) < 0 imply
,{l-0)/2n
9
Re8. = 1 (Cv) > (1 - 0)d{tl,/2(l
+ A„(l - W O -
= - - - /$2 n
4
P r o o f . Let K be a totally imaginary number field of degree 2n > 2. Assume t h a t
CK(P) < 0 f ° r some 0 satisfying \ < /? < 1. According to the proof of [Lou2, P r o p
A], Hecke's integral representations of Dedekind zeta functions yield
R e s s = 1 ( C x ) > (1 - /3)d{£-1)/2fn(P)(l
- 27rnd-/3/2n).
where fn(0) — P(^(P)/(2~)l3~1)n
is positive and log-convex in the range 0 > 0,
hence convex in the same range. Since fn(l)
= ( / n / / n ) ( l ) = ~ A n , we obtain
/n(/3) > / n ( l ) + (P~ 1)/„(1) = 1 + A„(l - 0) in the range 0 < 0 < 1. .
Gérard Boutteaux and Stéphane Louboutin
To get the term (1 —p)dK~
as large as possible, we would like t o be allowed
to choose /? = 1 - 2 / l o g d / c , and (K(l-(2/\ogdK))
< 0 would imply Res s = 1 (C/<) >
(2 + o(l))/\ogdK
where o(l) is an explicit error term which approaches to zero as
dK goes to infinity. Discarding this error term, we would obtain (see the proofs
of Theorems 8 and 12): Let K be a sextic CM-field and assume t h a t (K(l ~(2/\ogdK))
< 0 . Then,
^'
(2)
^
WdK/dF
e7r 3 (logdF) 2 \ogdK '
and hK = 1 would imply dK < 2s 10 2 3 , pK := d](6 < 7500 and dF < 2 • 1 0 n . Since
we do not know how t o prove t h a t (,K(1 — (2/ \ogdK)) < 0 (which would hold true
if CK n a d no real zero in the range 1 — (2/\ogdK)
< s < l ) , we will have to be a
little more clever and we will obtain worse bounds (see Theorems 8 and 12 below).
Proposition 5.
(1) Let a > 0 be given and let K range over CM-field sextic fields.
(a) There exists da effective such that dK > da and Cx(l — (l/a\ogdK))
0 imply
(3)
Ress=1(C*) >
1
ae1/2a\ogdK
(b) For any c > 1, there exists datC effective such that dK
(l/a\ogdK)
< 0 <1 and C,K(j3) < 0 imply
(4)
<
Ress=1(C*)>
> da,c,
1
1-/3
cel/2a
•
(2) (See [Lou7]f If F is totally real cubic number field, then
Ress=1(CF)<^log2rfF,
(5)
and \ < 0 < 1 and (F(P) = 0 imply
(6)
Proof.
R e s ^ C F ^ - ^ l o g
3
^ -
According to Lemma 4, Cx(l ~ (1/alogd/*-)) < 0 implies
Res s==1 (Cx) >
Fa(dK)
logd*;
el/2a
where Fa(x) = (1 + ( A 3 / a l o g x ) ) ( l - 6ne1/6a/x1/6)
is clearly > 1 for x > da large
enough (notice t h a t A3 = 6.245278 • • •). In the same way, 1 - ( 1 / a logoff) < 0 < 1
and (K(/3) < 0 imply
Rfis.=i(0r) >
where Ga(x)
= 1 - 67re
1/6a
16
/x '
\r/^Ga(dK)
is > 1/c for x > da,c = ( f ^ f T V l 0 large enough. •
nal sextic CM-fields
Lemma 6. (See [LLO, Lemma 15](. Setcx = ( 3 + 2 v / 2 ) / 2 < 3 andc2 = ( 2 + v / 3 ) / 4 <
1. The Dedekind zeta function of a number field M has at most one real zero
in the range 1 — (1/ci logoff) < s < 1 and at most two real zeros in the range
1 - ( l / c 2 l o g d M ) < s < 1.
4. Non-normal sextic CM-fields with non-normal maximal totally
real real cubic subfields
From now on, we let K denote a non-normal sextic CM-field whose maximal totally
real subfield F = k is a non-normal cubic field. We write K = F(y/—d) where S is
a totally positive algebraic element of F. We let F denote the normal closure of F.
Hence, F is a real non-abelian normal sextic field with Galois group the dihedral
group of order six, and we let Lre denote the only quadratic subfield of F. Hence,
Ire = Q(Vdp)
and F = FLre = F(y/dp).
Recall t h a t there exists some integer
/ > 1 such t h a t dp — dLref2 and dF = d\JA
= dLreo9F (see [Mar]). In particular,
d | r e | d?F | dK and dF \ dp.
4.1. First case
We assume t h a t K contains an imaginary quadratic subfield Lim = Q ( v / ~ d L i m )
of discriminant —dLim < 0. Then WK = u)Lim, QK — QLim — 1 and K = FLim =
F(y/~dLim),
i.e. we can choose S £ Q . We set M = LreLim
and IV = FM. Hence,
M is an imaginary biquadratic bicyclic number field, N is a dihedral CM-field of
degree 12 and we have the following (incomplete) lattice of subfields:
J^- ^-t
Һ
1
6
2
1
- - ^ ŕ
ï
*
2 . J _ — Ĺ re _-—'M
--"'2
Lemma 7. Recall that we have set c\ = (3 +
2\f2)/2.
(1) Since CTV/CM = (C,K/C,Lim)21 any real zero of the entire function CK/CL^
is
at least a double zero of CN , hence is less than 1 — (1/ci logd/v). Moreover,
dN divides dK. Hence, (C,K / ^Lim)(s) > 0 for 1 - ( l / 3 c i \ogdK) < s <l.
(2) QK = I, WK — ^Lim and hLim divides hK. In particular, hK = l implies
hLtm = 1(3) Assume that hK = 1. Then, any real zero of CK is at least a double zero of
CAT- Hence, CK(S) < 0 in the range 1 — (l/3ci logd/c) < s < 1, by Lemma
6.
P r o o f . Since dN/dM = (dK/dLim)2
and since djj divides (dLredLim)2,
(use t h e
conductor-discriminant formula), we deduce t h a t dN | d?KdLc | dK. Since
K/Lim
is a cubic extension, then QK — QLim — 1 and /ijr,.m = ft£. divides hK (see [LOO,
Theorem 5]). Therefore, if hK = 1 then Ltm must be one of the nine imaginary
Gérard Boutteaux and Stéphane Louboutin
quadratic fields of class number one (i.e. dLim € { 4 , 8 , 3 , 7 , 1 1 , 1 9 , 4 3 , 6 7 , 1 6 3 } ) ,
which implies (Lim(s) < 0 for 0 < s < 1, and using Ov = (QK/C,Lim)2C,M, we obtain
the last assertion. •
Theorem 8. (Compare with Theorem 12 helow). Let K = FLim he a non-normal
sextic CM-field which is a compositum of a non-normal totally real cubic fi,eld F
and of an imaginary quadratic field Lxm.
Assume that CLim(s) < 0 in the range
0 < s < 1. Then,
cr\
h~ ^
(7)
H
Moreover,
\fdK/dF
* ~ Tu(\ogdF)HogdK
27
f r dK
°
1 6
hK = 1 implies dK < 2 • 10 , pK := d ^
-
10
19
•
< 34500 and dF < 3 • 10 1 3 .
P r o o f . Set c 3 = 37r 3 c 1 e 1 / 6 c i = 287.031 • • • (recall t h a t cx = (3 + 2 v / 2 ) / 2 ) .
first use (1) and the second point of Lemma 7 to obtain
,_
K
.
wK
[die
We
Res8=i({K)
~ STT^V dF R e s s = 1 ( C F ) '
We then use (3) with a = 3ci for which da = 10 1 9 (see the third point of Lemma
7) and we finally use (5). We obtain
c3(logdғ)2logcí
for dк
> Ю1'
As for the last assertion, we notice t h a t since NF/Q(TK/F)
1), we have dK > 3d?F which implies
> 3 (use Proposition
2V/W
K
~
c3(logdF)2 1og(34)
for dF > 3 • 10 1 3 , and dF < \/dK/3
K
which implies
8(3dK)1/4
> 1
-c3(log(rfA-/3))2logdK
for dK > 2 • 10 2 7 . •
By using Theorem 8, necessary conditions for class numbers of CM-fields to
be equal t o one (see [BL], [Oka] and [Bou]) and the technique developed in [LOO,
Sections 4.4 and 4.7] for computing class numbers of such non-normal sextic CMfields, we obtain:
Corollary 9. There are 134 non-normal
CM sextic fields K = FLim
with class
number one which are composita of a non-normal totally real cubic field F and of
an imaginary quadratic field LjTO, the ones given in TABLES 1, 2 and 3 below.
Non-normal sextic CM-fields
4.2. Second čase
We assume t h a t K = F(yf^ó) contains no imaginary q u a d r a t i c field (where 5 is
a totally positive algebraic integer of F ) , i.e. we assume t h a t we cannot choose
6 £ Z. We let ó\ — ó, 62 and 63 denote the three conjugates of <5 and set d =
M2Ó3 = NF/Q(6). We then set K t = F ( < / = ^ ) , K[ = F{yf^ďFTl) = F{y/-dLrJi),
Mi = F í x / 1 ^ " ) . Hence, M{/F is bicyclic biquadratic a n d Ki/F, F/F and ATJ/F
are t h e t h r e e q u a d r a t i c subextensions (with base field F) of t h e extension Mi/F.
We also set N = F (\/-<$i, >/-<$2, V~<53) and M 0 = F{sf^d) (recall t h a t F denotes
t h e normál closure of F). Hence, N is a normál CM-field with maximal totally reál
subfield N + — F (y/Si&2, y/^i&3) > N is the normál closure of K and the Mj's a r e
conjugate subfields of N. We háve the following incomplete lattice of subfields:
We will set M = Mi and K — K\.
Lemma 10. For 1 < i < 3, č/ie Mi are pairwise distinct but isomorphic.
Hence,
[N : F] = 8 and Gfd(NfF) = C2 x C2 x C2.
P r o o f . Assume t h a t two of t h e m , say M\ and M 2 are equal. T h e n for some OJ e F
2
2
2
we háve Ó2 = a ^ . Hence, d = <53a <5 a n d M 3 = F{\J—<S3) = F(%/—ď) is normál.
Since t h e Mj's are conjugate subfields of TV, we obtain M ~ M\ — M 2 — M 3 =
F{y/~^d) = F(y/dp, \f^-d). Since i ť / F is one of the t h r e e quadratic subextensions
of M/F and since if / F{\/dp) — F (for /í is totally imaginary), we obtain t h a t
K — F{y/~^ď) oř K = F(y/—ddp), contrary to t h e hypothesis t h a t K contains no
imaginary q u a d r a t i c subfield. •
Lemma 11. Recall that we háve set c2
factorization of Dedekind zeta functions:
(8)
(9)
(2 + \/3)/4.
Í[(CMJCF)
CN/CA
=
(CMO/CF)(CWC/0 3
We háve the following
It)
Gerard Boutteaux and St^phane Louboutin
(10)
CM/C F = (C/C/CF)(CK'/CF).
Therefore, any complex zero of the entire function CK /CF is at least a triple zero of
the Dedekind zeta function Cjv, hence is less than 1 - ( l / c 2 i o g d j v ) , by Lemma 6.
Moreover, dN divides d2K . Hence, ( C K / C F ) ( S ) > ® for 1 ~ ( l / 2 4 c 2 log df<) < $ < 1.
Proof.
Since G a l ( N / F ) is the elementary 2-group C2 x C2 x C 2 , using abelian
L—functions we easily obtain (8). To deduce (9), we notice t h a t ( M 3 = CM 2 = CMH
for the Mi are pair wise isomorphic for 1 < i < 3. In t h e same way, since MfF is
bicyclic biquadratic with quadratic subextensions KjF, K' JF and F / F , we obtain
(10). Let us finally prove t h a t dN divides o9K. To begin with, let / denote t h e
norm of the conductor of any one the quadratic extensions Mi/ F (since t h e Mt are
pairwise isomorphic for 1 < i < 3, we have f\ — / 2 = f$). Using the conductordiscriminant formula, we obtain t h a t dN divides d8p(fif2f-s)A
— d8-f12. Since
dM = d2pf', we obtain t h a t dN divides dxM/dlp.
Now, according to [Sta, Lemma
6], the different T>M/F divides t h e different VK/F.
Hence, dM —
d2pNM/Q(VM,F)
divides d2FNM/Q(VK/F)
= d2F(NK/Q(VK/F))2
= <PpdlK/dF.
Hence, dN divides
(d2pd2K/dF)12/dl?
= d ^ d ' ^ / d ^ 8 and noticing t h a t d F divides dF, we finally obtain
t h a t dN divides (dK/dF)24,
hence divides d2K. •
Theorem 12. (Compare with Theorem 8 above). Let K be a non-normal
CM-field with maximal totally real subfield a non-normal cubic field F and
that K contains no imaginary quadratic subfield. We have
sextic
assume
Moreover, /i£ = 1 implies dK < 2 • 10 2 9 , p A := d j / 6 < 76500 and d F < 3 • 10 1 4 .
P r o o f . Set c 4 = 127r 3 c 2 c 1 l 4 8 c a = 354.989 • • • (recall t h a t c 2 = (2 + v / 3 ) / 4 ) . Now,
there are two cases to consider.
First, assume t h a t CF has a real zero 0 in [1 — ( l / 2 4 c 2 \ogdK), 1[. Then, (K(0) —
0 < 0. Therefore (use (4) with a = 24c 2 and c = 48 for which d a>c = 6 • 10 7 ),
(12)
Res s = 1 (C/c) > ^ 1 7 ^ -
Using (1), (6), (12) and QKwK
^
\JdK/dF
fOT
-** > 6 • 10 7 .
> 2, we get
nq\
h-
(1J)
* * " 4^ei/48Calog3d7 ^ 27r3eil«c2(lOgd^-0oid^)
\/dK/dF
sfi
fOT
** ~
6
in7
' X°
(for d * > d F ) .
Second, assume t h a t CF has no real zero in [1 - ( l / 2 4 c 2 logcijf),l[. T h e n C F ( 1 ( l / 2 4 c 2 \ogdK)) < 0 and according t o Lemma 11, we conclude t h a t
CF-(l-(l/24c2logdA-))<0.
Non-norma! sextic CM-fields
Therefore (use (3) with a = 24c2 for which da = 10 2 2 ),
(14)
f
" « - ' « * > * 24c 2 evllo g d 7
° r dK "
l 22
° -
Using (1), (5), (14) and QK^K > 2, we get
(is)
\/dK/dF
Һ-K > \2ir
,„_.,
c e / _,
* ;дLľ,'f,
(log d ) „,__.,
íogd
3
1 4Sc
fотd
2
2
F
K
* >1022-
Since the right h a n d side of (13) is always greater t h a n or equal to the right hand
side of (15) (for C2 > 1/2), the lower bound (15) always holds.
Finally, the proof of the last assertion of this Theorem is similar to t h e proof
of t h e last assertion of Theorem 8. •
Lemma 1 3 . Assume that hK is odd, let Q be the only prime ideal of F which is
ramified in K/F (see Proposition 1) and let q > 2 be the rational prime in Q.
Assume that (q) = Q\Q\ is partially ramified in F/Q. Then, TK/F ~ Q\ for some
oddr>\.
P r o o f . Let a be any totally positive algebraic integer of F such t h a t K = F(\f~a).
2
There exists some integral ideal J of F such t h a t (4a) = l TK/k
(see L e m m a 2)
and TK/F = Q\ or Q2 for some odd r > 1 (see Proposition 3). Now, TK/F — Q\
r
2
r
would imply (4qr a) = J with J = TQ Q2.
Since the narrow class number of F is
odd (by Point 1 of Proposition 1), as in the proof of Proposition 3 we would obtain
r
2
r
:i
4q a = 7 for some 7 G F. Hence, K = F(y/—~a) = F(f—q )
= F(y/ q)
would
contain an imaginary quadratic subfield, a contradiction. •
Lemma 14. Let k be a number field of degree n > 1, let Ak denote its ring of
algebraic integers and let Q be a prime ideal of k above the rational prime q = 2.
Let e > 1 denote the ramification index of Q. If there exists a primitive
quadratic
character \o on the multiplicative group (Ak/Qr)*,
then 2 < r < 2e + 1, which
implies 2 < r < 2n + 1.
Proof. If m = z 3 i > o a * ^ t ' a * e (0,1} is the binary expansion of a non-negative
integer m > 0, then it is well know t h a t the 2-adic valuation jv 2 (m!) of ml is equal
to m — S(m) where S(m) = Yli>o ai- Hence,
^(7^TT2) = 5(m)-2m
rn
4 (m\y
(notice t h a t 5 ( 2 m ) = S(m)).
Let /:Q be the Q-adic completion of k and let UQ
denote t h e associated valuation. The Taylor series expansion
1 ( 2 ™) ! .,m
4™(m!) 2
is convergent if and only if
(2m)
u
xm
^.^
4 + o^Amí
o " w ; v 4mm\\2
( m ! ) 2 )~
=
i
l m_e(5(m)-2m)
' ~ ufc-^+oo
+ mí/ Q (x) = +00,
Gérard Boutteaux and Stéphane Louboutin
hence if and only if VQ(X) > 2e. Therefore, if a 6 Ak satisfies a = 1 (mod
Q2e+l),
there exists 0 € Ak such that a = /J2 (mod Q 2 e + 1 ) , which implies Xo(c*) = 4-1.
Since there must exist some a £ Ak satisfying a = 1 (mod Q r _ 1 ) and Xo(") 7^ 1 is
Xo is primitive, we do obtain that r must satisfy r - 1 < 2e -f- 1. Finally, since the
order of the group (Ak/Qr)*
must be even, we have r > 2. •
L e m m a 1 5 . / / (2) = Q] Q'2, is partially ramified in F/Q and if there exists a primitive quadratic character Xo on the multiplicative group (Ak/Q^)*,
then r = 5.
P r o o f . Since 2 < r < 5 and r is odd, it suffices to prove t h a t a quadratic character
on the multiplicative group (Ak/Ql)*
is not primitive. Since ker(Ak/Ql)*
—>
(Ak/Ql)*
= { ± 1 } , it suffices to prove that - 1 is a square in (Ak/Ql)*.
This
2
2
follows from the fact that if TX £ Q 2 \ Q , then - r e Q\\Q\,
hence ir' = 2 (mod Ql)
and (1 + vr)2 = - 1 (mod Ql). •
Theorem 16. Let K be a non normal sextic CM-field. Let F denote its totally real
cubic subfield. Assume that F is non-normal, that only one prime ideal Q of F is
ramified in the quadratic extension K/F and let q > 2 be the rational prime in Q.
(1) Assume that q > 2. Then, either
(a) (q) = Q i Q 2 Q 3 splits completely in F and TK/k - Q\> Q2 or Q3
(and the three corresponding sextic fields may be non-isamorphic
(e.eg.
cases 11 and 12 in Table 4 below)).
(b) (q) = Q i Q | is partially ramified in F and TK/k — 62(c) (q) = Qi Q 2 in F with fQx = 1 and / Q a = 2, then TK/k = Q\ •
(2) Assume that q = 2. Then, either
(a) (2) = Q1Q2Q3 splits completely in F and TK/k = Q\, Q 2 or Q\ (and
the three corresponding sextic fields may be
non-isomorphic).
(b) (2) = Q i Q | is partially ramified in F and Tj</k — Q\r
an
(c) (2) = Qi Q2 in F and TK/k — Q\ ° Ql ( d the two corresponding
sextic fields K are not isomorphic).
P r o o f . Use Proposition 3 and Lemmas 13 and 15. •
By using Theorem 12, necessary conditions for class numbers of CM-fields to be
equal to one (see [BL], [Oka] and [Bou]), by using Theorem 16 for constructing the
quadratic characters associated with the quadratic extensions K/F and by adapting
the technique developed in [LOO, Sections 4.4 and 4.7] for computing class numbers
of non-normal sextic CM-fields containing imaginary quadratic subfields, we obtain:
Corollary 17. There are 233 non-normal CM sextic fields K with class number one
which contain no imaginary quadratic subfield and whose maximal totally real cubic
subfields F are non-normal, the ones given in TABLES 4, 5 and 6 below.
Non-normal sextic CM-fields
5. Tables
TABLE 1. K = FLtm
case
1
2
3
^im
4
3
3
3
3
5
4
dғ
148
314
568
940
321
TABLE 2.
case
1
2
3
4
5
6
7
8
3
3
3
3
2
3
3
3
9
3
8
8
1
! 10
11
12
13
14
15
16
17
18
|
dьim
3
3
3
3
3
3
3
dғ
621
756
837
1620
1944
2241
2700
2808
3132
4104
4860
and (q) = Q is inert in F
dL
37
316
568
940
/
Pғ(x)
2
1
1
1
x3
x3
x3
x3
321
1
x 3 + xг - 4x - 1
+
+
-
xг
x2
x2
7x
-
Зx - 1
4x - 2
6x - 2
4
dк
24 • З3 372
2 4 . 3З . 7 9 2
24
pк
9.1611.79-•• 1
2 6 • З 3 - 7 1 2 14.34-••
• З 3 • 5 2 • 4 7 216.96- •
2 6 • З 2 - 107 2 13.69-
K^FL im a n d (q) — Q 3 is totaìly гamified in F
dL
69
21
93
5
24
249
12
312
348
456
60
165
732
921
5940
6588
8289
9153
11880
12744
20385
113
1320
1416
2265
148
404
564
37
101
141
19
20
21
22
23
24
4
4
4
4
4
4
756
1524
21
381
3124
781
25
8
148
26
11
1573
/
3
6
3
18
9
4
15
3
3
3
9
6
3
3
9
3
3
3
2
2
2
6
Pғ(x)
x3
x3
x3
x3
x3
x3
x3
x3
3
- 6x - 3
- 6x - 2
- 6x - 1
- 1 2 x - 14
- 9x - 6
- 9x - 5
-15x-20
- 9x - 2
x
3
x
x3
x3
x3
-
18x
18x
18x
12x
15x
-
x3
x3
x3
x3
x3
-
21x 21x27x 21x ЗЗx -
20
16
12
6
16
12
4
34
30
48
xл + x 2 - Зx - 1
x 3 - x 2 - 5x - 1
x 3 + x 2 - 5x - 3
2
2
x 3 - 6x - 2
x 3 + x 2 - 7x - 1
x 3 - l б x - 12
37
2
x 3 + xl
143
11
- Зx - 1
x 3 + x 2 - 7x - 2
dк
pк 1
10.24
Зг -232
T
2
2 • 3 - 7 10.94
З 7 • З l 2 11.31
2 4 • З 9 • 5 2 14.10
26 - З 1 1
14.98
З 7 • 8 3 2 15.71
4
7
4
2 • З • 5 16.72
1694
26 - З 7 - 132
4
7
2
2 • З - 2 9 17.57
6
7
2
2 • З • 1 9 19.22
2 4 - З 1 1 • 5 2 20.34
4
21.74
2 •37-52-ll2
24 - З 7 - 6 1 2
22.51
7
2
З • 3 0 7 24.30
9
2
З • 1 1 3 25.12
26 • 3 7 - 5 2 1 1 2
27.40
2 б • З 7 • 5 9 2 28.05
З 7 • 5 2 • 1 5 1 2 32.80
4
8
2
2 • 37
8.39
2 8 • 1 0 1 2 11.73
8
2
2
2 • З • 4 7 13.11
2 8 • З 6 • 7 2 14.46
2 8 • З 2 • 1 2 7 2 18.26
8
2 • l l 2 - 7 1 2 23.20
21X-372
11.87
î l 5 • 132
1 7 . 3 4 - ••
i and Stéphane Louboutin
T A B L E 3. K " =
~ i
dLlrn
3 '
3
FL, „ a n d (q) = QiQІ
is p a r t i a ł l y ramified in F
dy
321
564
dL
321
f
1
Pғ(x)
x 3 + xl - 4x - 1
x3
x3
x3
x3
3
4
3
3
993
1101
993
1101
57
6
7
8
3
3
3
1524
2505
3.144
381
2505
3144
2
1
1
x3 + x 2 - 7 x - l
x3-x2-10x-5
x 3 - x 2 - lбx - 8
9
10
11
12
3
3
3
3
3252
3540
3576
3873
813
885
3576
2
2
1
x 3 + x 2 - 9x - 3
x3-x2-15x-15
x3 } x2 - 1 5 x - 3
13
14
15
16
3
3
4692
4764
3873
1173
4764
1
2
1
x3-x2-16x-17
x3-x2-17x-3
x' 4 - x 2 - - 12x - 6
3
3
3
6120
7032
7464
1605
7032
7464
2
1
1
x 3 + x 2 - 25x - 55
c 3 + x 2 - 14x - 1 8
x 3 - x 2 - 24x - 24
18
19
20
21
22
3
3
3
8220
8472
9192
8220
8472
9192
3
3
10200
10641
23
24
3
3
10812
12216
408
10641
10812
1
1
í
5
1
1
x
3
x
x3
3
x
x3
x8
25
26
27
28
29
30
3
3
14520
16689
3
3
3
10860
18201
19020
12216
120
16689
16860
18201
19020
3
20073
31
32
4
4
316
940
33
34
4
4
1708
1772
35
36
37
38
4
4
39
4
4
2636
2700
3596
-1364
4844
40
41
4
4
5356
7084
42
43
4
4
7244
7404
1 Ţ
1
]1
1
+
+
+
-
x 2 - 5x x2
6x .r 2 - 9x
x 2 - 8x -
2
1
1
5
3
+x
2
- x
+ x2
2
+ x
+ x2
- x2
2
3
3
12
3
-20x- 27x - ISx - 23x - 22x - 23x -
12
33
30
27
16
9
x 3 - x 2 - 35x - 57
3
x - ЗЗx - 22
x 3 + x 2 - 26x - 24
20073
1
1
1
1
x3
x3
3
x
x3
+
+
+
-
x2
x2
2
x
x2
316
940
1708
1772
І
1
1
1
xA
x3
x3
3
x
+
-
x 2 - 4x - 2
7x - 4
x 2 - 8x ~ 2
14x - 12
2636
12
3596
4364
4844
1
15
1
1
1
5356
7084
1
]
7244
7404
1
x 3 + x 2 - 20x - 38
1
x 3 + x 2 - 12x - 6
-
16x
ЗOx
20.т
ЗOx
-
10
48
30
24
x3-x2-16x-18
x3-15x-20
3
x - l l x - 8
x 3 + x 2 - 19x - 27
3
x -1 x 2 - 12x - 14
3
x - 2 2 * - 28
3
2
x + x - 1 9 x - l l
dк
З 3 • 107 2
24 - З 3 • 472
З3 -ЗЗl2
З 3 3672
З 3 • 54 192
24 • З 3 • 1272
З 3 - 5 2 • 167 2
26 - З 3 • 1312
24 • З 3 • 2 7 1 2
24 З 3 • 52 • 592
26
Pк
8.22992
11.9812.4013.51 13.82
16.31
17.59
17.7918.3018.3618.85-
З 3 • 149 2
З3•12912
2 4 • З 3 • 17 2 2 3 2 20.10
24 • З 3 • 3972 2 0 2 0 -
2
2 4 • З 3 • 5 2 • 10722.32
•
2 6 • З 3 • 2 9 3 2 23.00
fi
3
2
2 З • Зll
23.46 •
2
2 4 • З 3 • 5 2 • 137 24.23
•
3
2
t • З • 353' 24.48
3
2
6
2 • З - 383
25.152 6 • З 3 • 5 4 • 17 226.04 •
З 3 • 3547 2 2 6 . 4 1 2
2 4 • З 3 • 17 2 • 5 3 26.55
•
2 6 • З 3 • 5 0 9 2 27.65 •
6
3
2
4
2 -З -5 • l l
29 29
З 3 • 5 5 6 3 2 30.69 4
3
2
2
2
З -5 -281
30.79 •
3
2
З •6067
31.594
3
2
2
2 - З - 5 -317
32.05 3
2
3 -6691
32.63
«
.
ąг
8.582
7
2 6 • 5 Э • 4 7 2 12.3426-72 612
15.06 •
2 6 • 4 4 3 2 15.24 6
2
2
659
26 • З 6 • 54
2 6 29 2 - З l 2
26 • 10912
2б . ? 2 , 1 ? 3 2
174017.54 •
19.30
20.58 •
21.31 •
6
2
2
2 • 1 3 • 1 0 3 22.04 •
2
2
2
24.192 -7
l l -23
2 6 • 1 8 1 1 2 24.37 •
6
2 6 • З 2 • 6 1 7 2 24.55-
l;.»
case
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
(ІLim
4
-1
1
4
4
4
4
4
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8
8
11
11
11
11
11
dғ
8556
9676
11884
14316
14956
16044
18604
21324
469
756
2177
2233
4193
4641
5089
5369
6153
6601
7665
8113
8505
9905
10353
14385
16737
20545
25249
30849
568
1304
1944
3736
8920
11032
11608
14296
14680
28504
473
2024
2101
2233
5368
dt
8556
9676
11884
14316
14956
16044
18604
21324
469
21
2177
2333
4193
4641
5089
5369
6153
6601
7665
8113
105
9905
10353
14385
16737
20545
25249
30849
568
1304
24
3736
8920
11032
11608
14296
14680
28504
473
2024
2101
2233
5368
TABLE 3 (continued)
f Pғ(x)
dк
pi
J
2
Ь
2
2
1 x + x - 27x - 51
2 • З 23" • 31
25.76 • •
3
6
2
2
1 x - 22x - 12
2 41 - 59
26.84 - •
3
H
2
1 x - 19x - 24
2 • 2971
28 753
2
6
2
2
1 x - x - 27x - 21
2 • З • 1193
30.59 3
2
6
2
1 x + x - 32x - 6
2 • 3739
31.04 ••
3
2
б
2
2
2
1 x + x - 36x - 54
2 З • 7 • 191
31.77 • •
3
2
6
2
1 x - x - 36x - 18
2 • 4651
33.38
3
2
6
2
2
1 x - x - 44x - 84
2 З • 1777
34.93 •
3
2
Л
2
1 x + x - 5x - 4
7 67
10.74 • •
3
4
3
6 x -6x-2
2 • 3° 7
12.591 x 3 + x 2 - 8x - 5
7 3 - З П 2 17.92
3
2
3
2
2
1 x + x - 8x - 1
7 -ll -29
18.07 ••
3
2
3
2
1 x - x - 12x - 7
7 599
22.30 •
3
2
2
3
2
2
1 x f x - 14x - 21
З • 7 13 17 23.07 •
3
2
3
2
1 x - x - 14x - 11
7 • 727
23.78 ••
3
3
2
2
1 x - 17x - 23
7 13 • 59
24 21 • •
3
2
2
3
2
1 x -x -12x-3
З • 7 • 293
25.343
3
2
2
1 x - IЗx - 9
7 • 23 • 4 1
25.94 •
3
2
2
3
2
1 x - 27x -- 19
З • 5 • 7 73 27.27 • •
1 x 3 - lЗx - 5
7 3 • 19 2 61 2 27.79 • •
9 x 3 - 27x - 51
З 1 0 • 5 2 7 3 28.23
1 x 3 - 17x - 19
5 2 7 3 283 2 29.70
3
2
2
3
2
2
1 x + x -22x-43
З • 7 • 17 • 29 30141 x 3 - x 2 - 36x - 69
З 2 - 5 2 • 7 3 • 137 2 33.63 •
1 x 3 - x 2 - 36x - 27
З 2 7 3 797 2 35.37 • •
1 x 3 + x 2 - 40x - 67
5 2 7 3 • 587 2 37.88 • •
3
1 x - 19x - 9
7 3 • 3607 2 40.57 • •
1 x 3 + x 2 - 42x - 45 З 2 • 7 3 13 2 113 2 43.37
3
l
2
1 x - x - 6x - 2
Ѓ • 71
11.71
1 x 3 - x 2 - llx - 1
2 9 • 163 2 15.45 3
9
10
9 x - 9x - 6
2 • З
17.651 x 3 + x 2 - 14x - 22
2 9 • 467 2 21.943
9
2
2
1 x - 22x - 16
2 • 5 • 223
29.32 • •
3
2
9
2
2
1 x + x - 14x - 10
2 7 197 31.481 x 3 - x 2 - 23x - 5
2 9 1451 2 32.02 • •
3
2
9
2
1 x + x - 24x - 4
2 1787
34.32 • •
1 x 3 + x 2 - ЗOx - 70
2 9 • 5 2 • 367 2 34.62 •
3
9
2
2
1 x - 34x - 40
2 • 7 • 509
4319 ••
1 xл - 5x - 1
ll 3 • 43 2 11.613
2
6
3
2
1 x - x •- Ю x - 6
2 • ll 23 18.86 • •
1 x3-x2-llx-8
ll 3 191 2 19.10
1 x 3 + x 2 - 8x - 1
7 2 - l l 3 - 2 9 2 19.49 •
3
2
6
3
2
1 x + x - 19x - 23
2 • ll • 61
26.11 •
Uerard lioutteaux
i Louboutin
TABLE 3 (continued).
casє
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
dLim
11
11
11
11
11
19
19
19
19
19
19
19
43
43
43
67
163
dғ
7084
dL
7084
/1
1
1
1
1
57 5
1425
1
3021
Ш. i
3496
1
З3'ł.,
43 04
456 ЗГ
5529
5529
1
1
5624
5624
39064 39064
1
473
473 1
1
7224
7224
1
11137
11137
469
469 1
1304
1
1304
15961
17116
27049
37609
15961
17116
27049
37609
Pғ(x)
x3
x3
x3
x3
x3
4- x 2
- x2
4- x 2
4- x 2
4- x 2
x3
x3
x3
x3
x3
- x 2 - 8x - 3
4- x 2 - 9x - 6
- lЗx - 14
- 18x - 16
+ x 2 - 20x - 39
- 19x
- ЗOx
— 16x
- 34x
- 34x
- 11
- 32
— 2
- 56
- 32
x 3 - x 2 - 24x - 28
x 3 - x 2 - 46x - 26
x 3 - 5x - 1
x 3 4- x 2 - 19x - 7
x 3 4- x 2 - 22x - 8
x 3 4- xl - 5x - 4
x3 - x
l
- llx - 1
dк
PK
• 2 3 2 28.64 •
2
37.54 •
ll - 1451
2 4 • l l 3 • 3 8 9 2 38.43l l 3 -2459 2
44.76ll 3 • 1 3 2 • 2 6 3 2 49.96 •
24 -72 • П
3
3
З2
З2 •
26 •
26
З2 •
54 • 193
193 • 5 3 2
19 3 • 2 3 2
Зб • 193
193 • 9 7 2
18.3823.61 •
24.7926.1528.886
3
2
2 • 1 9 • 3 7 29.042 6 • 1 9 3 • 2 5 7 2 55.42 •
l]2 433
14.582 6 - З 2 • 7 2 • 4 3 336.182
2
3
7 • 3 7 - 4 3 41.7972 • 673
2b 1633
15.6525.53-
Non-normal sextic CM-fields
A' does not contain any imaginary quadratic field
and (g) = Q1Q2Q3 splits in E
case
I
2
3
4
5
6
dғ
q
8
9
10
11
12
13
148
148
148
148
316
316
321
34!
321
404
404
404
404
107
139
491
691
211
307
59
79
163
43
179
179
283
14
15
16
17
469
564
568
621
263
331
83
151
18
19
733
733
127
127
20
21
22
23
24
756
785
993
211
23
139
31
7
25
26
27
28
29
1 30
1101
1345
1373
1373
1425
1524
1901
1901
71
71
43
19
31
31
Pк(X)
xb
x6
6
x
x6
ь
x
6
x
xb
6
x
x6
6
x
x6
x6
6
x
xb
b
x
x6
xb
xb
x6
b
x
b
x
b
x
b
x
b
x
xb
x6
b
x
6
x
b
x
x6
+ 49x 4 + 223x 2 + 107
+ 77x 4 + 235x 2 + 139
4
2
+ 89x + 1787x + 491
+ 65x 4 + 459x 2 + 691
4
2
+ 65x + 4 2 7 x + 211
4
2
+ 53x + 7 1 1 x + 3 0 7
+ Зlx 4 + 86x 2 + 59
4
2
+ 141x + 291x + 79
+ 27x 4 + 126x 2 + 163
4
2
+ 57x + 247x + 43
+ ЗЗx 4 + 235x 2 + 179
+ 1073x 4 + 1335x 2 + 179
4
2
+ 41x + 251x + 283
+ 20x 4 + 128x 2 + 263
4
2
+ ЗЗx + 291x + 331
+ ЗЗx 4 + 203x 2 + 83
4
+ 36x + 144x 2 + 151
+ 29x 4 + 163x'2 + 127
+ 80x 4 + 892x 2 + 127
4
2
+ 33x + 279x + 211
4
2
+ ЗOx + 97x + 23
4
2
+ 186x + 8505x + 139
4
2
+ 68x + 776x' + 31
4
2
+ 14x + 45x' + 7
+ 32x 4 + 252x 2 + 71
+ 32x 4 + 108x 2 + 71
4
2
+ 551x + 4 2 2 x + 4 3
4
2
+ 629x + 4763x + 19
4
2
+ 24x + 108x + 31
+ 93x 4 + 451x 2 + 31
dк
Pк |
2
37 • 107 11.52 •
2
37 • 139 12.03 •
2
37 •491 1 4 . 8 5 - •
2
37 • 691 15.72- ••
2
4
79 -211 16.61 •••
4
2
2
79 • 307 17.69-••
59 • 107 2 13.50-••
2
2
79 • 107 14.18 • • З
2
З 2 - 107 • 163 16.00
4
2
13.83- ••
2 -43- 101
2 4 • 1 0 1 2 • 179 1 7 . 5 4 - 4
2
2
1 0 1 • 179 17.54 •••
4
2
2 • 1 0 1 -283 18.94-••
7 2 • 6 7 2 263 19.66 •
4
2
2
2 • З • 4 7 • 331 21.73--2 6 7 1 2 • 83 17.29- ••
b
2
З
2 3 • 151 19.68 •••
2
127 • 7 3 3 20.21 •••
127 • 7 3 3 2 20.21 • ••
2 4 - З b - 7 2 -211 22.22 •••
5 2 • 23 • 1 5 7 2 15.55-••
2
2
З • 139 • З З l 22.70-••
2
2
З • 31 3 6 7 18.30--•
2
2
5 • 7 • 269 15-26-••
2
71 • 1373 22-61 •••
71 • 1373 2 22.61 •••
З 2 • 5 4 • 19 2 • 4321.06- ••
4
2
2
2 • З • 19 • 127 18.79-•2
21.95-••
31 •1901
31 • 1901 2 21.95-••
4
2
24
4
2
4
2
2
Gérard Boutteaux and Stéphane Louboutin
TABLE 4 (continued).
case
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
dғ
1944
2101
2228
2300
2557
2713
2857
3325
3356
3941
4409
6133
6133
7668
7753
7753
9076
10333
10077
11853
15061
15641
15700
16649
17929
20252
21281
21913
22229
23417
30169
30868
31165
32421
37437
g
139
23
19
107
23
3
19
23
107
71
11
7
7
19
3
19
3
7
31
7
7
2
3
2
3
11
2
68
46996
51316
85300
3
7
2
3
3
7
7
7
3
3
3
69
126664
3
70
142876
3
63
64
65
66
67
Pк{X)
pк
dк
x + 21x + lllx 2 + 139
2Ь • 3 1 U 139 28.40- ••
b
4
2
2
2
x + 60x + 128x + 23
l l • 23 • 1 9 1 21.59 •
x b + 8 1 x 4 + 7 2 7 x 2 + 19
2 4 • 19 5 5 7 2 21.334
x b + 749x 4 + 567x 2 + 107
2 - 5 4 • 2 3 2 • 107 28.76- ••
x b + lбx 4 + 60x 2 + 23
23 • 2 5 5 7 2
23.06
16.74-••
x b + ЗЗx 4 + 155x 2 + 3
3-27132
b
4
2
2
x + 222x + 137x + 19
23.17-••
19 • 2 8 5 7
b
4
2
4
2
2
x + 116x + 2 6 7 2 x + 23
5 - 7 • 1 9 23 25.17- • •
x b + 89x 4 + 1659x 2 + 107
2 4 • 107 8 3 9 2 32.61
b
4
2
x + 20x + 96x + 71
7 2 • 71 5 6 3 2 32.14
x b + 467x 4 + 6 9 0 x 2 + 11
11 • 4 4 0 9 2 24.45- ••
x b + 68x 4 + 328x 2 + 7
25.31 •• •
7-61332
6
4
2
x + 229x + 83x + 7
25.31
7-61332
b
4
2
4
ь
2
x + 1473x + 363x + 19
32.31 •••
2 -З - 19-71
ь
4
2
2
x + 129x + 59x' + 3
3 • 7 7 5 3 23.76- ••
x b + ЗOx 4 + 129x 2 + 19
19- 7 7 5 3 2 32.33 •
x b + 1017x 4 + 143411x2 + 3
2 4 • 3 • 2 2 6 9 2 25.05- ••
x b + 36x 4 + 96x 2 + 7
7 • Ю З З З 2 30.12x b + 36189732x4
З 2 • 31 • 3 3 5 9 2 38.28 • - •
2
+28035372408x + 31
x b + 3141x 4 + 627x 2 + 7
З b - 7 • 4 3 9 2 31.53-••
xb + 1 2 x 4 + 3 2 x 2 + 7
7 • 15061 2 " 34.15 x b + 72x 4 + 37x 2 + 2
2Л • 15641 2 35.36 •••
x b + 81x 4 + 107x 2 + 3
2 4 3 ' 5 4 • 1 5 7 2 30.07-••
x b + 2316144x4 + 404773789x2 + 2
2A • 16649 2 36.11 •••
b
4
2
x + 2090x + 7069x + 3
31.43- ••
3•179292
x b + 377x 4 + 3 5 2 5 9 x 2 + 11
2 4 11 - 6 1 2 - 8 3 2 40.64 •
b
4
2
6
2
2
x + 33528x + 565x + 2
2 • 1 3 • 1637 39.19 •
x b + 230x 4 + 1194ІX2 + 3
3 • 1 7 2 • 1289 2 33.60 •
x b + 340x 4 + 18784x2 + 7
7•222202
38.88 • • •
b
4
2
x + 32x + 45x + 2
2Ó • 23417 2 4 0 . 4 5 - •
b
4
2
2
x + 19955x + 598x + 3
3•30169
37.38- ••
2 4 • 3 • 7 7 1 7 2 37.67- ••
x b + 569x 4 + 1327x2 + 3
b
4
2
2
2
2
x + 2277x + 76371x + 7
43.52 •
5 - 7 - 2 3 -271
x b + 48x 4 + 180x2 + 7
З 2 -7 • 101 2 1 0 7 2 44.10- ••
x b + 556749204x4 + 136272x2 + 7
З 2 -7 • 12479 2 46.26-••
x b + 346798ІX4 + 3660259x2 + 3
2 4 - З - З l 2 3 7 9 2 43.33 •
4
2
xb + 1 3 3 6 1 x 4 + 9 0 7 x 2 + 3
2 -зiгsгэ 44.62-••
b
4
x + 310760391641x
2 4 - 3 - 5 4 - 8 5 3 2 52.86-••
+33331797907x 2 + 3
x b + 82841249297885x4
2 b • 3 • 7 1 2 • 2 2 3 260.31 • • •
+30290200633872943x2 + 3
b
4
2
x + 9749333x + 106807x + 3
2 4 • 3 • 2 3 2 • 1553 2 62.78-••
b
4
fa
Non-normal sextic CM-fields
case
1
2
3
4
5
6
7
8
9
10
11
12
13
гu~
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
1
42
dғ
148
148
148
148
148
148
148
316
316
316
321
321
321
404
494
404
469
469
469
469
469
473
473
564
564
568
621
621
621
733
733
733
837
837
993
1101
1300
1300
1304
1345
1373
1373
Ч
23
31
227
331
467
499
547
23
131
547
23
31
307
7
67
347
2
11
47
223
239
71
139
163
211
139
19
47
103
2
11
71
2
79
7
223
11
67
11
19
3
151
K does not contain any imaginary quadratic field
and (g) = Q1Q2 in E
Pк(X)
dк
Pк I
2" - 23 • 37^ 8.92 • xь + 25x4 + 51x2 +23
6
4
2
4
2
9.37
x + 29x + 67x + 31
2 • 31 37
1
2
2' • 37 - 227 13.06 x 6 + 25x4 + 175x2 + 227
x 6 + ЗЗx4 + 227x2 + 331
2 4 • 37 2 • 331 13.91•
6
4
2
x + 49x + 615x + 467
2 4 • 37 2 • 467 14.73x 6 + ЗЗx4 + 299x2 + 499
2 4 • 37 2 • 499 14.89x 6 + 41x4 + 319x2 + 547
2 4 • 37 2 • 547 15.122 4 23 • 79 2 11.48x b + 25x4 + 79x'ź + 23
6
4
2
4
2
x + 21x + 119x + 131
2 • 79 • 131 15.34
6
4
2
2 4 - 79 2 • 547 Ï9.47
x + 113x + 1195x + 547
ь
4
2
2
x + 37x + IЗlx' + 23
З • 23 • 107 2 11.54
2
2
x 6 + 38x4 + 89x2 + 31
З • 31 • 107 12.13
6
4
2
2
2
x + 81x +315x +307
З • 107 • 307 17.78
x ь + 9x4 + 19x2 + 7
2 4 -7 • 101 2 10.22
4 .
,
2
x 6 + 25x4 + 79x2 + 67
14.89
6
4
2
4
2
x + 33x + 211x + 347
2 • 101 • 347 19.59
b
4
J
b
l
г
x + 9x + 14x + 4
2 - 7 • Ь7 15.53
x 6 + 36x4 + 168x2 + ll
7 2 • 11 -67 2 11.58
x 6 + 20x4 + 104x2 + 47
7 2 • 47 • 67 2 14.75
x 6 + 29x4 + 195x2 + 223
7 2 • 67 2 • 223 19.13
x 6 + 24x4 + 164x2 + 239
7 2 • 67 2 • 239 19.35
xb + ЗOx4 + 229x2 + 71
11* -43 2 -.71 15.85
x 6 + Зlx4 + 202x2 + 139
ll 2 43 2 • 139 17.73
xb + 57x4 + 435x2 + 163
2 4 • З 2 • 47 2 • 16319.31
x 6 + 21x4 + 123x2 + 211
2 4 -З 2 -47 2 -211 20.15
b
4
2
2 b • 71 2 • 139 18.84
x + 53x + 463x + 139
x b + ЗЗx4 + 75x2 + 19
З b • 19 • 23^13.93
З 6 • 23 2 • 47 16.20
x 6 + 45x4 + 483x2 + 47
З 6 • 23 2 • 103 18.47
x 6 + 60x4 + 504x2 + 103
b
4
2
2Л • 7 3 3 2 12.75
x + 15x + llx + 2
11 -733 2 13.44
x 6 + 32x4 + 188x2 + ll
71 • 733 2 18.34
x 6 + 37x4 + 339x2 + 71
xb + 27x4 + 15x" + 2
2Л -З b Ъ\l 13.32
x 6 +45x 4 + 195x2 + 79
З 6 •Зl 2 • 79 19.52
xb + llx4 + 26x'" + 7
З 2 -7-331 2 13.79
b
4
2
2
x + 288x + 564x' + 223
З • 223 • 367 2 25.42
x b + 13x4+43x'2 + ll
2 4 - 5 4 -11 13 2 16.27
x 6 + 49x4 + 587x2 + 67
2 4 • 5 4 • 13 2 • 6721.99
2 b • 11 • 163 2 16.29
xb + 41x4 + 75x2 + 11
xь + 42x4 + 65x" + 19
5 2 • 19 • 269 2 18.03
b
4
2
x + 60x + 136x + 3
3 1373 2 13.34
6
4
2
!
x + 92x + 1664x + 151
151 •1373 2 25.64-••
2
б 7
1 Q 1
• 1
20
Gérard Boutteaux and Stéphane Louboutin
TABLE 5 (continued).
case.
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
dғ
1573
q
Pк(X)
dк
Pк
l l 4 • 1 3 2 • 71 2 3 . 6 6 -
xь
4853
71
31
79
3
11
67
2
59
11
19
19
2
2
3
2
4933
4933
7
19
xь + 37x 4 + 275x 2 + 7
x 6 + 113x4 + 123x 2 + 19
60
61
62
63
64
65
66
67
68
5081
8837
3
31
3
3
7
11
3
3
23
69
70
8909
8909
2
7
9293
2
11
2
7
3
7
7
7
3
3
19
3
7
31
58
59
71
72
73
74
75
76
77
78
79
80
81
82
83
84
2101
2589
2636
2708
2808
3021
3356
3508
3604
3736
3957
4765
4841
5685
6185
6401
6557
7028
7796
8372
9413
9749
9805
9812
9813
9869
10261
10292
10721
12092
12269
12309
12333
+ 20x 4 + 104x 2 + 71
x b + 40x 4 + 356x 2 + 31
l l 2 -31 1 9 1 2
22.70З 2 • 79 • 8 6 3 2 28.44 • • •
x" + 24x 4 + 84x 2 + 79
4
2
2 4 • 3 • 6 5 9 2 16.58 • • •
x° + 193x + 59x + 3
x b + 17x 4 + 51x 2 + ll
2 4 • 11 - 6 7 7 2 20.78-••
x b + 8 1 x 4 + 171x 2 + 6 7
x° + 28x 4 + 17x 2 + 2
2 b • 3° • 1 3 2 • 6728.43 •••
2Л • З 2 • 1 9 2 • 5 3 220.44 • • •
x° + 29x 4 + 183x 2 + 59
2 4 - 59 • 8 3 9 2 29.54- •
2 4 • 11 - 8 7 7 2 22.65 •••
x b + 13x 4 + 27x 2 + ll
x
b
4
2
+ 13x + 35x + 19
2
4
• 1 7 2 • 19 • 5 3 2 25.04 •••
x° + 9 7 x 4 + 1739x2 + 19
2°
x D + 93x 4 + 258x 2 + 4
19 4 6 7 2
25.34 •••
2 b • З 2 • 1 3 1 9 2 3163
•
x b + 79x 4 + 1087x2 + 2
2Л • 5 2 • 9 5 3 2 23.79
x° + 46x 4 + 29x 2 + 3
3 • 4 7 2 - 103 2 20.31 •••
x° + 191x 4 + 207x 2 + 2
b
4
2 J - 2 3 2 • 211" 23.94 • 7•49332
19 4 9 3 3 2
2
23.54 •
27.80- -
+ 14x + 25x + 3
3 • 508V
x ь + 93x 4 + 195x 2 + 31
З 2 • 5 2 • 31 379 2 31.63
x
x° + 10x 4 + 17x 2 + 3
x b + 257x 4 + 59x 2 + 3
3 • 3 7 2 • 173 2 22.29-
x b + 36x 4 + 312x 2 + 7
ò
x
+65x
4
+399x
2
2
+ ll
2
4
24
x b + 13x 4 + 35x 2 + 3
x b + llx 4 + 15x 2 + 2
x 6 + 1792x4 + 1444x2 + 7
x
2Л • 9293 2 29.73 • • •
+ ЗOOx + 97x + 2
11 • 9 4 1 3 2
x° + 21317x4 + 46437ІX2 + 7
x° + 407569x4 + 5075623ІX2 + 3
Ъl •7 • 3 7 2 • 5 3 2 29.60
2 4 - 3 - 11'- - 2 2 3 2
x b + Ю l x 4 + 83x 2 + 7
x
+ 1044x + 5424x' + 7
7 • З l 2 3 3 1 " 30.05
+ 473x 4 + 143x 2 + 3
4
24 - З З l 2
2
x° + llx + 2 6 x + 3
3-71
x b + 5226ІX4 + 280087x2 + 19
x
832
• 151
3•122692
x b + 60x 4 + 768x 2 + 7
4
2
З2 -7- 112-3732
2
+ 62820x + 69024x + 31
-
26.12-•
2
26.48 • • •
2 4 • 19 • 3 0 2 3 2 37.49 •••
xb+97x4 + 155x2+3
b
25.71 •••
З 2 • 7 • 3 2 7 1 2 29.61 • •
7 - 7 1 2 - 139 2 29.66-
2
x b + 20x 4 + 24x 2 + 7
b
31.48-••
26 • 9749 2 30.21 ••
x b + 99x 4 + Зlx 2 + 2
x
34.86 • •
2à • 59 2 • ІЬЃ 29.31
7 - 5 9 2 • 1 5 1 2 28.67-••
2
4
23.81 •••
23 • 8 8 3 7 2
x D + 1 0 5 x 4 + 139x 2 + ll
n
3•19492
2
2 4 • 3 • 7 2 • 1 3 2 • 2 324.38
• • •
x b + ЗЗбx 4 + 7540x 2 + 23
4
2
7 • 7 9 • 8 3 25.88 • • •
- 7 2 • 11 -251 2 28.56 •••
x ь + 49x 4 + 395x'2 + 3
ь
20.64 • - •
3 - 5 2 • 1237 2 22.04- ••
З
2
-31-4111
2
27.69-•
31.93 •
40.94 • • •
-nal sextic CM-fields
TABLE 5 (continued).
case
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
dғ
12401
12788
14165
14229
14420
15212
15252
15284
16532
16721
17165
17273
17684
17717
17780
19112
19544
19869
21853
23612
24884
26612
30356
33885
34172
38204
38420
39704
41240
43368
44072
45884
51404
56264
58904
88700
93260
Я
3
11
23
7
11
3
19
3
3
3
23
3
11
2
3
3
11
7
7
3
3
3
3
7
11
3
3
3
3
19
11
3
3
3
3
3
3
Pк{X)
dк
pк
3 124012 27.79- ••
xb + 194x 4 + 245x 2 + 3
x b + 2997x 4 + 6 0 1 9 x 2 + 11
2 4 • 11 -23 2 •139 2 34.87 • • •
b
4
2
2
x + 147536x + 1835892x + 23
5 • 23 • 2833 2 40.80-••
x b + 12x 4 + 24x 2 + 7
З b • 7 • 1 7 2 • З l 233.51 •••
x b + 185x4 + 123x 2 + 11
2 4 • 5 2 - 7 J l l 1 0 3 2 36.29- ••
x b + ЗЗlЗx 4 + 203x 2 + 3
2 4 • 3 • 3 8 0 3 2 29.75-••
2
2
2
x b + 208953x4 + 11619x2 + 19
2 4 • З • 19 • З l • 4 140.51 • • •
b
4
2
24-3-38212
29.80-••
x + 41x + 4 1 9 x + 3
b
4
2
4
2
30.59- x + 17x + 23x + 3
2 -3 4133
3 • 232 • 7272 30.70-•
x b + 83x 4 + 1322x' + 3
b
4
2
2
2
x + 2336x + 176292x + 23
5
23 • 3 4 3 3 43.50-••
3 • 2 3 2 • 7 5 Ґ 2 31.04-x b + 94x 4 + 53x 2 + 3
2 4 11 -4421 2 38.85--x b + 641x 4 + 291x" + 11
2Л • 7 2 • 2 5 3 1 2 36.86--x b + 15x 4 + 5 5 x 2 + 2
4
2
2
2
31.34-••
xb+61x4+35x2+3
2 - 3 - 5 - 7 -127
2 b • 3 • 23892 32.10 •
x є + 29x 4 + 47x 2 + 3
b
4
2
6
2
x + 149x + 5 1 0 3 x + ll
2 • 7^-11 -349 40.17- ••
З 2 -7-37 2 179 2 3 7 . 4 6 x ь + 69x 4 + 1011x2 + 7
7 13 2 -41 4 38.66 • 1
x b + 236x 4 + 8944x 2 + 7
b
4
2
2 4 • 3 • 59032 34.45- ••
x + 209x + 107x + 3
4
b
4
2
2 -3-6221 2 35.06-••
x + 113x + 2 5 0 7 x + 3
x b + 565x 4 + 3035x 2 + 3
2 4 • 3 • 6653 2 3 5 . 8 5 - - !
2 4 • 3 7589 2 37.46 •••
x b + 3025x 4 + 56423x2 + 3
З b - 5 2 - 7 - 2 5 1 2 44.75--x ь + 24x 4 + 132x 2 + 7
2 4 • 11 • 8543 2 48.39- ••
x b + 1145x4 + 699x 2 + ll
b
4
2
2 4 -3 9551 2 40.44 •••
x + 186193x + 2027x + 3
2
2
2
b
4
2
4
x + 113x +2891x + 3
2 •3-5 17 • 113 40.52 •••
2 b • 3 • 7 2 • 709 240.96-••
x b + 34513x 4 + 11627x2 + 3
2 6 - 3 - 5 2 Ю З l 2 41.49-••
xь + 59166ІX4 + 19373647295x2 + 3
2
2
2
57.39--x b + 39750177x4 + 618651147x2 + 192 b •З 13 19- 139
2 ь - 7 2 - П - 7 8 7 2 52.67- •
x b + 7957605x4 + 2047039x2 + 11
4
2
2 -ЗП471
42.99 - • •
x b + 14833x4 + 765899x2 + 3
4
2
2
44.65 • • •
x b + 659338673x4 + 49539876107x2 + 3 2 -3-71 181
b
2
2
ö
4
2
2 • 3•13 • 541 46.01 •••
x + 788594737x + 20558987x + 3
b
2
2
b
4
2 • 3 • 37 • 199 46.72
• • •
x + 2225x + 342539a:* + 3
2 4 • 3 • 5 4 • 887 253.55 • • •
x b + 127921x4 + 43787x^ + 3
2 4 • 3 • 5 2 • 4663 254.46 • • •
x b + 121861x4 + 2454407x2 -ř 3
22
TABLE 6. K
case
dғ
1
568
2
1101
2024
3
4
2233
2505
5
2589
6
3124
7
3252
8
3596
9
3941
10
11
4844
12
5901
13
6153
14
6237
8637
15
8745
16
17
10868
18
12660
14189
19
15252
20
21
16116
22
17889
23
19113
24
22425
25
23028
26
29553
27
31101
28
32073
34152
29
36872
30
31
37176
32
38580
41172
33
34
41865
47004
35
52764
36
37
53736
38
73176
39
76200
40
98844
41
108325
42 201129
Gérard Boutteaux and Stéphane Louboutin
does not contain any imaginary quadratic field and (q) = Qi21 in E
2
3
2
11
3
3
11
3
2
7
2
7
3
7
3
3
11
3
7
3
3
3
3
3
3
3
7
3
3
11
3
3
3
3
3
3
3
3
3
3
7
3
dк
Pк
Pк(X)
n . 7lг
xь + 7x 4 + 10x'2 + 2
14.752
b
4
2
J
2
12.40x + 8x + 12x' + 3
3 • 367
2 1 1 • lľ 2 • 23 2 22.53x b + llx 4 + 3 0 x 2 + 2
x b + 74x 4 +77x' 2 + ll
7 2 • lľ* 29 2 19.49 •
3 J • 5 2 • 167 2 16.31
xb + l l x 4 + 3 0 x 2 + 3
З 3 • 863 2 16.49
x b + lбx 4 + 36x'2 + 3
2 4 • 11 J - 71 2 21.80x b + 193x 4 + 1859x 2 + 11
2 4 -3 J -271 2 17.79x b + 281x 4 +3183x 2 .+ 3
x ь + 9x 4 + lбx'2 + 2
2 9 -29 2 -Зl 2 27.29
x b + 37x 4 + 147x 2 + 7
7 J • 563 2 21.84ь
4
2
x + 65x + 24x + 2
2 9 • 7 2 • 173 2 30.14b
4
2
2
x + 452x + 2744x' + 7
З • 7 J • 281 2 24.99x b + 10x 4 + 21x'2 + 3
3 J • 7'2 • 293 2 22.00З в - 7 J - ll'2 25.45x b + 96x 4 + 252x 2 + 7
З d • 2879'2 24.64x b + 1804x4 + 612x' 2 +3
3
2
2
2
З 5 lľ -53
24.74 •
x b + 130x 4 + 117x'2 + 3
2 4 • ll 3 13 2 • 19 2 33.03
x b + 117x 4 + 979x 2 + 11
b
4
2
4
J
2
2
x + 25x + 147x + 3
2 -3 - 5 -211
27.987Л • 2027 z 33.48 •
x b + 229x 4 + 11347x2 + 7
4
J
2
2
b
4
2
2 -3 -Зl -41
29.78 •
x + 25x + 5 1 x + 3
x b + 25297x 4 + 47595x2 + 3
2 4 • 3 J • 17 2 • 79 230.333 J • 67 2 • 89'2 31.40x b + 19x 4 + 90x 2 + 3
x b + 175x4 + 4 0 2 6 x 2 + 3
3 J • 23 2 • 277 2 32.10
x b + 47671x4 + 822x 2 + 3
3 J • 5 4 • 13 2 • 23 23386b
4
2
4
x + 49x + 267x' + 3
2 -3 J • 19 2 • 101 2 34.16
x b + 19x 4 + 18x 2 + 3
3 J -9851 2 37.12b
4
2
2
J
2
x + 1857804x + 471072x + 7
З • 7 • 1481 43.49
x b + 322x 4 + 23073x'2 + 3
3 J Ю 6 9 1 2 38.152 b • 3 J • 1423 2 38.96 •
x b + 145x4 + 171x 2 + 3
b
4
2
b
J
2
x + 229059609x + 855080347x + 11
2 П -419
49.63 •
b
4
2
b
x + 375539437x + 32882926911x + 3
2 • З á • 15492 40.08 •
b
4
2
4
J
2
2
x + 241x + 75x + 3
2 • 3 • 5 • 643 40.57x b + 29329x4 + 1047339x2 + 3
2 4 • 3 J • 47 2 • 73 241.46b
4
2
x + 102958x + 30321x + 3
3J • 5 2 • 2791 2 41.702 4 - 3 J -3917 2 43.34xь + 433x 4 + 267x 2 + 3
2 4 • 3 J • 4397 2 45.04x° + 28549x4 + 3128295x2 + 3
fo
4
2
2 b • З 3 • 2239 2 45.31x + 56317x + 86415x + 3
x b + 2386728ІX4 + 2424100779x'2 + 3
2 b • 3 J • 3049'2 50.23 •
x b + бlx 4 + 207x 2 + 3
2 b • 3 J • 5 4 • 127 250.91 •
2 4 • 3 J • 8237 2 55.52cb + 2293x 4 + 6999x 2 + 3
x b + 12962664x4 + 309216012x'2 + 7
5 4 • 7 J • 619 2 65.93
x b + 137137x4 + 7947x'2 + 3
3 J • 67043 2 70.36 •
Non-normal sextic CM-fields
23
References
[BL]
[Bou]
[Lan]
[LLO]
[LO]
[LOO]
[Loul]
[Lou2]
[LouЗ]
[Lou4]
[Lou5]
[Louб]
[Lou7]
[LYK]
[Mar]
[Oka]
[Sta]
[Wa]
G. Boutteaux and S. Louboutin. T h e ciass number one problem for some non-normal
sextic CM-fields. Analytic number theory (Beijing/Kjoto, 1999), 27-37, Dev. Math. 6,
Kluwer Acad. Publ., Dordrecht, 2002.
G. Boutteaux. Détermination des corps à multiplication compiexe, sextiques, non galoisiens et principaux. PҺD Thesis, in preparation.
S. Lang. Algebraic Number Theory. Spгinger-Verlag, Grad. Texts Math. 110, Second Edition.
F . Lemmermeyer, S. Louboutin and R. Okazaki. T h e class number one problem for sorne
non-abeìian normal CM-fields of degree 24. Journal de Théorie des Nombres de Bordeaux
1 1 (1999), 387-406.
S. Louboutin and R. Okazaki. Determination of all non-normal quartic CM-fieids and of aìl
non-abelian normal octic CM-fields with class number one. Acta Arith. 6 7 (1994), 47-62.
S. Louboutin, R. Okazaki and M. Olivier. The class number one probiem for some nonabeiian normal CM-fields. Trans. Amer. Math. Soc. 3 4 9 (1997), 3657-3678.
S. Louboutin. On the class number one problem for nonnormal quartic CM-fields. Tôhoku
Math. J. 4 6 (1994), 1-12.
S. Louboutin. Lower bounds for rełative class numbers of CM-fields. Proc. Amer. Math.
Soc. 1 2 0 (1994), 425-434.
S. Louboutin. Determination of all quaternion octic CM-fields with class number 2. J.
London. Math. Soc. (2) 5 4 (1996), 227-238.
S. Louboutin. Computation of relative class numbersof CM-fieids. Math. Comp. 6 6 (1997),
173-184.
S. Louboutin. Upper bounds on | L ( l , x ) | and applications. Canad. J. Math. 5 0 (1999),
794-815.
S. Louboutin. Explicit bounds for residues of Dedekind zєta functions, values of Lfunctions at s — 1, and reiative class numbers. J. Number Theory 8 5 (2000), 263-282.
S. Louboutin. Expiicit upper bounds for residues of Dedekind zeta functions and values
of L-functions at s = 1, and expłicit îower bounds for relative class numbers of CM-fields.
Canad. J. Math. 5 3 (2001), 1194-1222.
S. Louboutin, Y.-S. Yang and S.-H. Kwon. T h e non-normal quartic CM-fields and t h e
dihedrai octic CM-fiełds with ideal class groups of exponent < 2. Preprint (2000).
J. Martinet. Sur ľarithmétique des extensions à groupe de Galois diédral ď o r d r e 2p. Ann.
Inst. Fourier (GrenoЫe) 19 (1969), 1-80.
R. Okazaki. Non-normal class number one problem and the least prime power-гesidue. In
Number Theory and Applications (series: Develoments in Mathematics
Volume 2), edited
by S. Kanemitsu
and K. Györy from Kluwer Academic Publishers (1999) pp. 273-289.
H.M. Stark. Some effective cases of the Brauer-Siegel theorem. Invent. Math. 2 3 (1974),
135-152.
L.C. Washington. Introduction
to Cyclotomic Fields. Grad. Texts M a t h . 8 3 , SpringerVerîag.
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E-maü address: l o u b o u t i Ф i m l . u n i v - m r s . f r
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