MULTIPLICITIES OF DIHEDRAL DISCRIMINANTS

MATHEMATICSOF COMPUTATION
VOLUME58, NUMBER 198
APRIL 1992,PAGES 831-847

MULTIPLICITIES OF DIHEDRAL DISCRIMINANTS

DANIELC. MAYER
Dedicated to the memory of Helmut Hasse

Abstract. Given the discriminant dk of a quadratic field k, the number
of cyclic relative extensions N\k of fixed odd prime degree p with dihedral
absolute Galois group of order 2p , which share a common conductor /, is
called the multiplicity of the dihedral discriminant d^ = f2^p~^d^ . In this
paper, general formulas for multiplicities of dihedral discriminants are derived
by analyzing the p-rank of the ring class group mod / of k . For the special
case p = 3 , c4 = -3 , an elementary proof is given additionally. The theory
is illustrated by a discussion of all known discriminants of multiplicity > 5 of
totally real and complex cubic fields.

Introduction
Let p be an odd prime and AT|Q a cyclic extension of degree p. Then it
is well known [9, 15, 7] that the conductor of K must have the form / =
Pe 'Qi •■■It> where e = 0 or e = 2, /> 0, and the q¡ are pairwise distinct
rational primes satisfying q¡ = X(mod p) for i = X, ... , t. The discriminant
of K is just a power of the conductor, d¡( = fp~x . If, for any positive integer
/, the number of cyclic extensions K\Q of degree p which share the same
conductor / is denoted by m(f), then

/-i/ p
where p = dimF,(Qx(/)/Q* -qx(f)p) = dimr,(SyL, U(Z/fZ) ®Zp¥p) = t + w
with ®x(f) = {r£®x \(r,f) = X}, Q; = {reQx \r= X(mod*/)}, and
w = \e. Moebius inversion yields an explicit formula for m(f):

m(pe-qx---qt) = (p-l),+w-1 .

It is the aim of the present paper to establish similar formulas for multiplici-
ties of discriminants in the case of non-Galois extensions L|Q of degree p with
dihedral normal closure yV of degree 2p. For the sake of illustration, the for-
mulas are applied to discriminants with multiplicities up to 16 of non-Galois

Received December 10, 1990; revised April 5, 1991.
1991MathematicsSubjectClassificationP. rimary 11R20,11R11,11R16.
Key words and phrases. Dihedral fields, quadratic ring class groups, cubic fields.
Research supported by the Austrian Science Foundation, Project Nr. J0497-PHY.

©1992 American Mathematical Society
0025-5718/92 $1.00+ $.25 per page

831

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832 D. C. MAYER

cubic fields which occur in the most extensive recent numerical tables [10, 5,

11].

1. Cubic discriminants

Assume that k is a fixed quadratic field with discriminant dk . If N\k is
a cyclic cubic relative extension with conductor / and with absolute Galois
group Gal(/V|Q) ~ S3, the symmetric group on three symbols, then [8, p. 578]
/ must have the form

/ = 3e • qx ■■■qt

with 0 < e < 2, t > 0, and pairwise distinct rational primes q¡ ^ 3 satis-
fying c7,= (^) (mod 3) for i = 1, ... , t. Furthermore, the 3-exponent e
is restricted to the values 0, 2 if dk = ±1 (mod 3), and to the values 0, 1
if dk = 3 (mod 9), but no restrictions arise for dk = -3 (mod 9). An in-
teger / of this form will be called an admissible conductor for the quadratic
discriminant dk.

For any positive integer /, define the multiplicity m(dk , f) of f with re-
spect to dk to be the number of nonconjugate cubic fields L\Q with coinciding
discriminant dE = f2 -dk . Then the following general formula for the recur-
sive determination of multiplicities of cubic discriminants holds. This result
will be proved in a more general context in §3.

Theorem 1.1. Let f = 3e •qx ■■•qt be an admissible conductor for the quadratic
field k with discriminant dk and 3-class rank p. Then

£m(<4,/') = 5(3'+mü-á-i) ,

/'I/

where the sum runs over all divisors f of f. Here,

(0 ife = 0,
w = I 2 ife = 2 anddk = -3 (mod 9),

V 1 otherwise,

and Ô = dimF3(4,3(/)/4)3(/)n(Qx(/)rc;/cx(/)3)), where k*(f) (resp.

Qx(f)) denotes the numbers in kx (resp. Qx) which are coprime to f,kf =

{y £ kx I y = X(mod x /)} is the group of generators of the ray mod f of k,
and Ikii(f) = Ik,!^ kx(f) with the group Ik 3 of generators a £ kx of all

principal ideals acfk which are cubes of ideals of k .

2. Pure cubic discriminants

First, the complete solution of the multiplicity problem for the special case
of pure cubic discriminants is obtained in a totally elementary way with the aid
of the following well-known relationship [4] between the normalized radicand
D = m-n2 of a pure cubic field L = <Q(\/D), where m > n > 0 are squarefree
coprime integers, and the conductor / of the corresponding relative extension

N\k,

( 3mn if D £ ±1 (mod 9) (field of Dedekind's 1st kind),
S ~ I mn if D = ±1 (mod 9) (fieldof Dedekind's2nd kind).

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multiplicities of dihedral discriminants 833

Theorem 2.1. Let f = 3e •qx■■■qt > X bean admissible conductor for the special
quadratic discriminant dk = -3, i.e., 0 < e? < 2, t > 0, and q¡ ^ 3 are
arbitrary rational primes for i = X, ... , t. Put

u = #{X<i<t\q¡ = ±X (mod 9)},
v = #{1 < / < 11q¡ = ±2, ±4 (mod 9)}.

Then the multiplicity m(f) := m(-3,f) of the pure cubic discriminant
-3 •f2 is given by ife = 2,

(2l

m(3e-qx---qt) = \ 2" ■Xv if e = X,

l2".X„_, ife = 0,

where X¡ = \(V + (-X)J+X)for all j >-X.
Moreover, the multiplicities of conductors with 3-exponents e = 0, 1 satisfy

the relation
m(qx---q,) + m(3-qx---qi) = 2'-x.

Proof. To see that all claimed conductors are really admissible for dk = -3,
observe that dk = -3 (mod 9) and that the condition q = (&■)(mod 3) is
satisfied by every prime q .

First, the case e = 2 is treated separately. The relation f = 32 •qx ■■■qt is
equivalent to / = 3mn, 3 | mn, and thus also to D = 0 (mod 3). In this
case, there are 2t+x choices for the exponent systems 1 < w0, wx, ... , wt < 2
in cubefree radicands D = 3W°•q™]■■■q™' which all share the same value of
mn = 3-qx---qt. But only one of the two systems (iu0, ... ,wt) and (3 -
w0, ... , 3 - wt) belongs to a normalized radicand. Hence,

m(32 •qx■■■q,) = \2t+x = 2'.

Second, the cases e = X and e = 0 are investigated simultaneously. The

relation f = 3-qx---qt is equivalent to / = 3m«, 3 \ mn, and further to

D = ±2, ±4 (mod 9), whereas f = qx---qt is equivalent to / = mn, 3 f

mn, and also to D = ±1 (mod 9). In both cases, there are 2' choices for

exponents l<Wi,...,w(<2in cubefree radicands D = q™'■■■q™' which all

share the same value of mn = qx■■q,, but some of them (D = ±1 (mod 9))

give rise to conductor / = mn and the others (D = ±2, ±4 (mod 9)) to

conductor / = 3mn. Again, only one of the two systems (wx, ... , wt) and

(3-wx, ... , 3 - w,) belongs to a normalized radicand. (Both, the normalized

and the nonnormalized radicand, are of the same Dedekind kind.) Therefore,

m(qx ■■■qt)+ m(3-qx--- qt) = \2l = 2'~x.

To separate these two multiplicities, it is convenient to fix a value u > 0 of
the number of prime divisors q = ±1 (mod 9) of D and to argue by induction
with respect to the number v > 0 of prime divisors q = ±2, ±4 (mod 9) of
D. Obviously, u + v = t.

Induction start, v = 0, 1 :
In the case v = 0, we have mn = qx--qu with u > X and D = ±X
(mod 9), whence

y_,:= An(c?1--c7u) = 2"-1,

To := m(3 -qx---qu) = 0.

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834 D. C. MAYER

In the case v = X, we have mn = qx ■■■qu •qu+x and D = ±2, ±4 (mod 9),
whence

m(qx ■■■qu ■qu+i) = 0=Y0,

Yx:= m(3 • qx ■■■qu - qu+x) = 2".

Induction step, v -» v + X:
If the new prime factor qu+v+x= ±2, ±4 (mod 9) and its square are mul-

tiplied by a radicand D = ±1 (mod 9), then there are generated two new
radicands D-q™^{ = ±2, ±4 (mod 9) with 1 < wu+v+x < 2. However, if
they are multiplied by a radicand D = ±2, ±4 (mod 9), then one of the two
new radicands is congruent ±1 (mod 9) (the one, where q^+^+l represents the
square of D in the group U(Z/9Z)/{±X} ~ C(3)) and the other is congruent
±2, ±4 (mod 9). Thus,

m(Qi ■■■Qu+v+i) = m(3 ■qx ■■■qu+v) =: Yv,

Yv+x := m(3 -qx-- qu+v+i) = m(3 -qx--- qu+v) + 2 • m(qx ■■■qu+v)

= yv + 2y„_i.

Consequently, the numbers Y¡ (j > -X) satisfy a binary linear recursion,

Yj+x= Yj+ 2Ty_i for j > 0, with initial values T_i = 2U~Xand Y0= 0. This

recursion can be solved by diagonalization of the corresponding matrix M in

the equation

m-(\ o)-(ä)-

The characteristic polynomial of M is (denoting by / the identity matrix)

det(x • / - M) = x2 - x - 2 = (x + 1) •(x - 2).
The eigenspaces of M with respect to the eigenvalues -1 and 2 are

ker(M+ I) = ®- (}A , ker(M-2-I) = Q- (2A ,

and M becomes diagonal under the transformation

Therefore, the solution of the recursion can be represented in the form

(/>,) =*«■(rny-T-A'-T-'-(rl)

Hence, the solution is Yj = 2U-Xj with X¡ := \(V + (-X)i+X) for all ; >
-1. D

Remark. The proof of Theorem 2.1 used a very special elementary technique
which cannot be applied to other types of cubic fields. In the next section, this
result will be rederived as a particular instance of a much more general formula,
which is deduced by completely different methods, using ring class groups of
quadratic number fields.

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multiplicities of dihedral discriminants 835

Examples. As an application, the minimal occurrences of some higher multi-

plicities of pure cubic discriminants are determined. The first three cases appear

in tables of complex cubic fields [1, 5, 12] which are ordered by discriminants.

The further ones are constructed by means of Theorem 2.1, taking the smallest

possible prime factors. These examples show that the normalized radicands as-

sociated with a fixed discriminant of higher multiplicity are spread rather widely

in a table of pure cubic fields which is ordered by radicands, such as [11].

1. m(f) = 2 for / = 32-2, dE = -912, occurs in [1], with associated
radicands D = 6, 12, and e = 2, u = 0, v = 1.

2. m(f) = 4 for / = 32•2 • 5 , dE = -24 300, occurs in [12], with associ-
ated radicands D = 30, 60, 90, 150, and e = 2, u = 0, v = 2.

3. m(f) = 3 for / = 3-2-5-7, dL = -132300, occurs in [5], with
associated radicands D = 10, 140, 490, and e = X, u = 0, v = 3.

4. m(f) = 8 for / = 32-2-5-7, dL = -X 190700, occurs in [11], with
associated radicands D = 210,420,630, 1050, 1260, 1470,2100,
2940,and e = 2, w= 0, v = 3.

5. m(f) = 5 for / = 3-2-5-7-11, dL = -16008300, occurs in [11],
with associatedradicands D = 770, 3850, 7700, 10780, 16940, and
e = 1, u = 0, v = 4.

6. m(f) = 6 for /= 3-2-5-7-17, dL = -38234 700, occurs in [11],
with associated radicands D = 1190,2380,8330,11900,20230,
40460, and e = X, u = X, v = 3.

1. However, m(f) = 1 will never occur, since 7 is not a member of

the sequence (Xj)j>_x in Theorem 2.1. The same is true for m(f) =
9, 13, 14, 15.
8. m(f) = 16 for /= 32-2-5-7-11, dL = -144074700, occurs in [11],

with associated radicands D = 2310,4620,6930,11550,13860,
16170, 23100, 25410, 32340, 34650, 48510, 50820, 69300, 76230,
80850, 97020,and e = 2, u = 0, v = 4.
9. m(f) =11 for / = 3 •2 •5•7 • 11• 13, dL = -2 705402700, with as-
sociated radicands D = 10010,20020,70070,100100,110110,
260260, 350350, 550550, 650650, 700700, 910910, and e = 1, u =
0, v = 5.
10. m(f) = 10 for /= 3-2-5-7-11 ■17, dL = -4626398700, with
associated radicands D = 13090,65450, 130900, 183260,222530,
287980,458150,719950, 1007930,1112650, and e = X, u = X,
v = 4.
11. m(f) = 12 for / = 3-2-5-7-17-19, dL = -13802726 700, with
associated radicands D = 22610,45220, 158270,226100,384370,
429590,768740,791350, 859180, 1582700, 2690590,3007130,and
e = 1, u = 2, v = 3.

3. Dihedral discriminants

As before, assume that k is a fixed quadratic field with discriminant dk.
With p an odd rational prime, let N\k be a cyclic relative extension of degree
p with conductor / and absolute Galois group Gal(yV|Q) ~ Dp , the dihedral
group of order 2p . Then / is a rational integer and [2] must have the form

f = pe-qx-qt

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836 D. C MAYER

with O < e < 2, t > O, and pairwise distinct rational primes q¡ ± p sat-
isfying q¡ = (^) (mod p) for i = X, ... , t. Furthermore, the p-exponent
e is restricted to the values 0,2 if p is unramified in k, and to the values
0,1 if p | dk , except for the special configuration where p = 3 and dk = -3
(mod 9). An integer / of this form will be called a p-admissible conductor
for the quadratic discriminant dk .

For any positive integer /, define the p-multiplicity mp(dk , f) of f with
respect to dk to be the number of nonisomorphic fields L|Q of degree p shar-

ing the same discriminant dE = fp~x •d]p~x'' . By the translation theorem
of Galois theory, mp(dk, f) is also the number of cyclic extensions N\k of
degree p with absolute group Gal(/V|Q) ~ Dp sharing the common conductor

/-

Remark. If / is not a p-admissible conductor for dk , then certainly mp(dk, f)
= 0. But on the other hand, mp(dk , f) may be zero even for a p-admissible
conductor / for dk , as will be shown for infinite families of fields in Corollaries

3.2 and 3.3.
Denote by ^k(f) the group of (fractional) ideals of k coprime to /, and by

3lkj = {acfk \a£Qx(f)-kf} the so-called ring mod f of k, where Qx(/)

denotes the numbers in Qx which are coprime to /, and k? = {y £ kx \ y =. X

(mod x /)} is the group of generators of the ray mod f of k. Then, by the
Artin reciprocity law of class field theory [8], mp(dk , f) can be interpreted as
the number of subgroups %f of index p of Jk(f) which contain 3Hk,f but
not âêkj' for any proper divisor f'\f, f'^f- With the aid of this fact, the
following formula for the recursive determination of multiplicities of dihedral
discriminants can be derived.

Theorem 3.1. Let f = pe-q\-qt be a p-admissible conductor for a given
quadratic field k with discriminant dk and with p-class rank p = pk(p). Then

Ymp(dk,f') = -^—x(pp+,+w-ê-X),

f'\f

where the sum runs over all divisors f of f. Here,

( 0 ife = 0,
w = I 2 if e = 2, p = 3, dk = -3 (mod 9),

I 1 otherwise,

and Ô= 0(f) = dimfp(Ik,p(f)/Ik,p(f) n (®x(f)kxkx(f)p)), wherethe num-

bers in kx which are coprime to f are denoted by kx(f), and IktP(f) =
h,pri^x(f) with the group Ikp of generators a £ kx of all principal ideals
atfk which are pth powers of ideals of k.

Proof. Since the ring class group mod f of k, Jrk(f)lâêkj, is an abelian
group, its p-elementary subgroup is isomorphic to ^k(f)l(^k,f^k(f)p)-
Hence, any subgroup ^ of index p in J^(/) which contains «%,/ must
in fact be an intermediate group ¿%kj •^k(f )p < %f < Sk(f ). Therefore,

Y>"P(dk,f') = H^<^k(f)\(Sk(f):^)=P, ^k,f^k(f)p<^}

f\f

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MULTIPLICITIESOF DIHEDRALDISCRIMINANTS 837

is exactly the number of the hyperplanes (subspaces of codimension 1) of
the p-elementary ring class group mod f of k, viewed as a vector space
over the finite field ¥p. But this number equals -¡^¡(pf' - X), where p' =

dimYp(J?k(f)lâlkJ-JFk(fY) is the p-rankof J?k(f)I¿%kJ.

If ¿t°k(f) = &k ft<-ñc(f) denotes the principal ideals of k coprime to /,
then the factorization relation of elementary abelian p-groups,

Sk(f)/&k(f)-Sk(f)p
=-(Sk(f)/^k,f-jrk(f)p)/(<?>k(f).jrk(f)p/^kj-Sk(f)p)

is equivalent to a direct product relation

Mf)l&k,rJk{fY

^(Jrk(f)/&k(f)-sk(f)p)x(â*k(f).sk(f)p/œkJ.Sk(f)p),

where the first factor is isomorphic to the p-elementary class group of k. Fur-
ther, the homomorphism kx -» ¿Pk, «h acfk induces an isomorphism

&k(f)'Sk(fY/&kJ.jrk(fy~k*(f)/(Q*(f).kf.Ik>p(f)).

Finally, it is well known that the local description of the congruence relation
mod x / yields an isomorphism

kx(f)/Qx(f)-kx

* (kx(f)/kx ) I (Qx(f)/qx(f) nkx)c U(cfk/fcfkI) U(Z/fZ)

and that the p-elementary subgroup of U(cfk/fcfk) / U(Z/fZ), which is isomor-

phic to kx(f)/Qx(f)-kx-kx(f)p , has p-rank equal t + w [2, 6, 8], whence

p' = p + t + w-S with

á = dimFp(/,)í)(/)/4,p(/)n(Qx(/)./:/x./cx(/)'')). D

Remarks. X. For totally real dihedral fields, a further decomposition of the

essential index ps into two parts is useful for practical purposes:

P3 = (Ik,p(f):h,P(f)n(Qx(f)kxUkkx(f)p)).(Uk:Ukn(qx(f)kxkx(f)p)),

where the first part involves only the principal plh powers of ideals which rep-
resent p generating classes of order p of k, and the second part concerns
exclusively the fundamental unit of k . Here, Uk denotes the unit group of k .
2. The proof of Theorem 3.1 and the previous remark show that generally
à(f) < min(p, t + w) (in particular, 6(f) = 0 when p = 0) in the cases

dk < -3 or p > 5, dk = -3, and 0(f) < min(p + X, t + w) in the cases
dk > 0 or p = 3, dk = -3 .

Note. It should be pointed out that the proof of Theorem 3.1 does not use the
full ray class group mod f of k but only the ring class group mod f of k,
which is obtained from the former by factoring out the part invariant under
the generating automorphism of Gal(/c|Q) [6, 8]. Otherwise, there would be
counted all cyclic extensions N\k of degree p sharing the common conductor
/, those with group Gal(/V|Q) ~ Dp as well as those with group Gal(yV|Q) ~
C(2) x C(p) (composita of k with cyclic extensions K\Q of degree p).

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838 D. C MAYER

The only case where the sum in the formula of Theorem 3.1 consists of a
single term is the unramified case f = X, which was treated in [8, p. 581] and

[14] already.

Corollary 3.1. The number of unramified cyclic extensions N\k of degree p is

mp(dk,X) = ^-r(p»-X).

Proof. If / = 1, that is, t = 0 and w = 0, then we have 0(f) = 0, since
6(f) < min(p + X, t + w) = 0, or also simply since k? = k* . D

In all other cases, single multiplicities can be obtained by Moebius inversion.

Theorem 3.2 (General multiplicity formula when w < X). Assume that f =
pe •qx•••qt > X with w < X is a p-admissible conductor for the quadratic field

k, put qt+x = pe when w = X, and define âmax = ma\{ô(qil ■■■q¡s) \ 0 < s <
t + w, X< ix < ■•■< is < t + w}. Then

mp(dk, qx ■■■qt+w)

1 t+w-l
P"
(P-X)

+ Y(-x)'+w-s-ps Y -—yz\-L

s=0 Kii<—<is<t+w

Proof. Observe that, for fixed p and dk , the general formula in Theorem 3.1,

Ymp(dk,f') = -^-1(pp+t^-ô-X),

f'\f P~^X

can be viewed as the sum relation X)/'i/ m(f) = n(f) between two integer-
valued functions n(f) = ^-L.(/>/>+«/)+«<(/)-<*-(/X) ) and m(f') = mp(dk, f).
Hence, an application of the Moebius inversion formula yields an expression
for a single multiplicity,

(oi--at+w=)Y E m/*F^ \)•»(*,•••«<.)

s=0 l</,<-<i,</+ti) v q" q,s '

t+W ., . v

Sc==n0 l\<<¡i.]<<-.-.<.<i-sL<<tt+J-Win r \ y /
t+w
1 1
s=0 l<i\<—<is<t+w
1 ItT+UwJ y

-^tEí-1)^'"5 E l-

i=0 i<i\<—<i,<t+w

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MULTIPLICITIESOF DIHEDRALDISCRIMINANTS 839

Now the second double sum equals ££o (-l)'+u,-i • (t+sw)= 0 ; hence,

m(qx ■■■ql+w) = P-——--pXp ' p°n*x

t+W

Y(-x)t+w-s'Ps Y l

s=0 l<i,<--<is<t+w
t+w

+Y Y (-xy+w~s-ps'(pôm"~o{9i^^]- x)

s=0 l<ii<---<is<t+w

where the first double sum equals ¿Zlto (-l)t+w~s -ps • (t+sw)= (p-X)t+w ; thus,

m(iQi ■■■Qt+wx)= -—1t-Pp- p l
p- p>max

(p - X)'+w

t+W

+Y Y (-x),+w~s-ps .(p<5™*-<5(<7'v-1«)'S)

j=0 l<¡i<--<i3</+íii

This formula can be simplified further, since the indices for composed con-
ductors, a(qit ■■■qis) with 0 < s < t + w and 1 < i\ < ■■■< is < t + w , can be
determined from those for prime conductors, S(q¡) with X< i < t + w .

Corollary 3.2 (The special cases of ¿max = 0,1 when w < X). Let f =
Pe •Qi■■•Qt > 1 with w < X be a p-admissible conductor for the quadratic
field k, put qt+x = pe when w = X, suppose that <5max< 2, and put u =
#{ 1 < i < t + w | d(q¡) = 0}, i.e., v := t + w -u is the number of "bad" primes.

1 (The case without constraints from principal pfh ideal powers [8, p. 582]).
If u = t + w, then <5ma=x 0 and

mp(dk, qx---qt+w) =pp-(p-X vi+UI-l

2 (Restrictions caused by principal pin ideal powers). If0<u<t + w-X,
then ¿max= 1 and

mp(dk,qx---qt+w)=pp-(p-X)u.{-?- p•

In particular, mp(dk , qx ■■■qt+w) = 0 for v = X.

Proof. X.If u = t + w , then S(q¡) = 0 for all 1 < /' < t + w , and consequently
<5(<7•i•i•Qis)= 0 for all 0 < s < t + w and 1 < ix < ■■■< is < t + w , whence
¿max= 0. In this case, the formula of Theorem 3.2 immediately degenerates to

mp(dk,qx---qt+w)=p»-(p-X)<+w-x .

2. lfO<u<t + w-X, then 6(q¡) = X for some X< i < t + w , and thus

¿max= 1. In this case, it turns out that

#{ 1 < ii < ■■■< h < t + w | ö(qh ■■■qis)= 0}= (Us

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840 D. C. MAYER

for all 0 <s < t + w , whence

™P(dk, qx ■■■Qt+w)

X Cp-i)t+w-l

= nP .

t+w n<5max-<5(?i,---9is)
t+w-s t „s

P-1i=0 1<Í1<---<ÍS<Í+«)

= PP.- a,_1)í+»-ti+W+j^(_1)í+«-, ,v.^.Q-1.|z
1
P
(p_ !)'+«-!+(_i )»+»-«y (uY(-x)u-s'p
= P -.1

= p"---[1(p- xy+w-x + (-X)t+w-"(p- 1)"]

= pP.(p- X)u- ^—^--^-^- .D

The following array shows the multiplicities mp(dk, qx •••qt+w) m depen-
dence on the total number t + w > 0 of prime divisors of the conductor / > 1,
and on the number 0 < u < t + w of those "nice" primes which do not cause
restrictions, for the special case p = 3 and p = 0. For positive 3-rank p > 0
of k , the numbers must only be multiplied by 3P , provided that still ¿max< 2.

t+w u= 0 X

1 01
2 10 2
3 12 0 4
4 32408
5 5 6 4 8 0 16
6 11 10 12 8 16 0 32
7 21 22 20 24 16 32 0 64
43 42 44 40 48 32 64 0 128

Example 1. For pure cubic fields L = Q(\/rD), the formulas with w = e < X of
Theorem 2.1 in the previous section, which were proved by elementary methods,
can be reobtained here as the special case p = 3, dk = -3, p = 0, and
u = #{1 < / < t + w | q,: = ±1 (mod 9)}, since for any X < i < t + w the
condition S(q¡) = 0 is equivalent to C3 € Qx(i7;)-/c9x •Ä:x(t7,)3 and hence to
q¡ = ±X (mod 9):

■)V+W-1 (-1) W+ÍD-1

w3(-3, qx---qt+w) = 2" ■

where v := t - u and v + w is the number of bad primes.

Finally, supplementary formulas must be established for the special case p =
3, dk = -3 (mod 9), w = 2.

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MULTIPLICITIESOF DIHEDRALDISCRIMINANTS 841

Theorem 3.3 (General multiplicity formula when w = 2 ). Let f = 32 •qx ■■■q¡
with w = 2 be a 3-admissible conductor for the quadratic field k. Then

mi(dk, Ï1 - Qi■■•Qt) Y i
-(32-,î(3 «V"«'»)_ 3l-<5(3-9,y?,J)\,_
i
= 3PY(~X-)'~S "3S

5=0 1<(|<—<<><(

Proof. An application of the Moebius inversion formula to the sum relation
"(/) = E/'|/ m(f) in Theorem 1.1, where n(f) = i(3'>+«/)+'"(/)-<5(/-> 1)

and m(f') = m3(dk,f), yields an expression for the single multiplicity of the
conductor / = 32 •qx■■■qt :

2t
m(32-qx---q=,Y) Y E ^ 32•qx■■■qt
y-Qw-Qi* n(3v-qii---qis) ,

v=0 s=0 l<ii<—<i,<t

where the Moebius function takes the values

0 ifv = 0,

(_l)«-*+i if v = i s
(-xy~s ifv = 2.

Consequently, (_l)'-^+l.I [ 1 -1
t
3<5(3-<7,v-?,s)
m(32-qx---q=t)Y Y
+(-\)t-s,L [y.y+2- 1 -1
5=0 1<¡|<-<1S<Í
i 2\ (' 3*(3*•«,••■*)
t
Y i(32_<5(3«i•■•*—>t)f-W'iii•■•«,)). o
= yY(-i)'~s'3s

í=0 !<»!<•••</,<<

Corollary 3.3 (The special cases of ¿max = 0,1 when w = 2). Let f =
32-qx--qt with w = 2 be a 3-admissible conductor for the quadratic field
k. Further, redefine ¿max = max{¿(3" • <?,,• • •qis) | 0 < v < 2, 0 < s < t, 1 <

ix < ■■■< is < t}, suppose that ¿max< 2, and put u = #{ 1 < i < t \ o(q¡) = 0} .

Then

m3(dk, 32-qx--q,)

3^+1-2< //¿(32) = 0, u = t,

= < 3^+1.2»+i -1(2'-"-' - (-1)'-"-1) z/¿(32) = 0, 0<u<t-X,

3^+i. 2". 1(2'-"-(-1)'-") i/«$(3) = 0, ¿(32) = 1,

I 3p-2' ifS(3) = X.

The first case is the unconstrained one [8, p. 582] where ¿max= 0, in the second
case the multiplicity is zero for u = t - X, in the third case for u = t, and the
fourth case is actually independent of u.

Proof. X. If ¿(32) = 0, u = t, then clearly ¿max = 0, and the formula in
Theorem 3.3 becomes

m3(dk, 32-qx■■■Qt)= 3" ¿ (-1)'- •3' - (f) • \(32 - 3) = 3P+X-2' .

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842 D. C. MAYER

2. However, if ¿(32) = 0 but 0 < u < t - X, then certainly ¿max = 1,
and S(32-qh ■■■qis)= ô(3-qir--qis) = ô(qh---qis) for all 0 < s < / and

1 < i'i < ••• < is < t. In this case,

m3(dk, 32-qx---qt)

= 3py^(-l)í_í-3J Y -(32-S{-i¡\-ih) - 3i-¿(*,-«->))

5=0 1<<1<—</,<<

= 3^(-!l)'-i-3î Y -i(3-X)-3x-5^-<¡'s)

5=0 l<ii<—</,<í

3" ¿(-l)t-'-3,.(')+(-ir«¿(-l)-'.3'.(").(3-l

1.55==00 ^' 5=0 ^'

= 3p-(2' + (-X)'-u-2u-2) .

3. In the case ¿(3) = 0, ¿(32) = 1, we have ¿(3-<7;i■••?,,) = S(qit •••#,,)
and ¿(32 •qit ■■■qis) = X for all 0 < 5 < t and 1 < z'i <■••</*< í, whence

¿max = 1 and

m3(4,32-c71---c?í)= 3"¿(-irí.3i Y \(3-3x-ô^-%))

5=0 l<ii<—<»,<!

=3«t(-iy-.3'.i(3-i).[Q-Q

¿(_l)'-«.3'.Q-(-l)'-»¿(-l)-'.3«-0)

L5=0

= 3^.(2' -(-1)'-"- 2") .

4. If, finally, ¿(3) = 1, then â(3 •qh ■■■q¡,) = X and ô(32 •qh ■■■qis) = 1 for

ail 0 < s < t and 1 < ix < • • • < is < t, whence ¿max = 1 and

t -'J).I(3-1) =3^2'. □

mi(dk,32-qx---qt) = yY(-iy~S'3

5=0

Example 2. For pure cubic fields L = Q(^D), the formula with w = e = 2
in Theorem 2.1 of the previous section can be reobtained as the special case

dk = -3, />= 0,and ¿(3) = 1:

m3(-3,32-qx---qt) = 2t .

The various formulas of §3 are illustrated in the Supplements section at the
end of this issue by a discussion of all known discriminants of multiplicity > 5
of totally real and complex cubic fields, which occur in the most extensive recent
numerical tables [10], [5].

Final Note. It should be particularly emphasized that all the calculations which
are necessary for the determination of the multiplicity mp(dk , f) of a dihedral

discriminant d^ = fp-^¿^X)l2 can be carried out entirely in the underlying
quadratic field k.

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MULTIPLICITIESOF DIHEDRAL DISCRIMINANTS 843

Given a quadratic field k with p-class rank p = pk(p) and discriminant
dk, and given a p-admissible conductor / for dk, we have to find p prin-
cipal pth powers ax, ... , ap of ideals prime to / which represent p gener-
ating classes of order p of the p-elementary class group of k , and addition-
ally a fundamental unit r\ of k if dk > 0 or dk = -3. Then Ik p(f) =
(ax,...,ap( ,r¡))-kx(f)p.

Next, we must examine successively, if n £Qx(f)-kf •kx(f)p and if

a, € (ax, ... , a,-,) • Uk-Qx(f)-kx-kx(f)p for i = X, ... , p,

whence we will be able to determine S(f') for all divisors f'\f and finally the

multiplicity mp(dk, f) of di = fp~xdkv~x^2, by means of Theorem 3.2 or
3.3.

In this manner it is possible to construct complete tables of complex or to-
tally real dihedral discriminants up to a given bound, |d¿| < B, and for a
fixed value of the prime p, by the computation of the p-ranks of ring class
groups mod / of quadratic fields k, varying the discriminants dk and the
p-admissible conductors / for each discriminant dk .

In particular, approximations can be determined for the asymptotic densities
of dihedral discriminants for various primes p > 5, similar to the densities
of Davenport and Heilbronn [3] for p = 3. As D. Shanks pointed out in his
review [13] of [1], the convergence of these approximations to the asymptotic
limits would be rather slow, because the really high multiplicities, which con-
tribute the essential part to the limit, unfortunately occur in very high ranges
of discriminants.

A drawback of the proposed method is that it does not seem to be suitable
for obtaining generating polynomials for the non-Galois subfields L of dihedral
fields.

Example. For any two positive integers m and B, let np(m, B) denote the

number of complex dihedral discriminants í/¿ = fp~xd(p~x'' of multiplicity

tnp(dk , f) = m , which are bounded by |í/¿| < B^p~x^2. (Observe that dihedral

discriminants are always complete ((p - l)/2)th powers.)

To make the ideas in the final note more concrete, we have computed the

303 968 quadratic discriminants in the range -106 < dk < 0 and determined

the class numbers and p-class ranks p = pk(p) of the corresponding imaginary

quadratic fields k for p = 3, 5, 7. With the aid of this information, we can

calculate the exact numbers np(m, B) for m = X,p-X,p+X, and for any

given upper bound B < 106.

(a) According to Corollary 3.1, the first component of np(X, B) is the

number of quadratic fields k with p-rank p = X and discriminant

\dk\ = \dL\2tO-»<B,

i.e., single unramified cyclic extensions (absolute class fields) N\k of degree p
with conductor f = X.

By Corollary 3.2,1, the second component is the number of ramified cyclic
extensions (ring class fields) N of degree p over quadratic fields k with p-rank
p = 0 (and thus S = 0) and discriminant

\dk\ = \dL/fp~x\2/{p-x)= \dL\2l(p-x)If2 < B/f2

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844 D. C. MAYER

with a single prime divisor qx of the conductor /, such that t + w = X. To
compute this number, we examine each of the relevant quadratic discriminants
dk for admissible prime conductors qx = ±1 (mod p), also taking into consid-
eration the possibility of p\f, w = X.

In the case of p = 3, we must further add the number of pure cubic dis-
criminants (dk = -3) of multiplicity 1, which can be evaluated by counting
primes and products of up to three primes below certain bounds, according to
Theorem 2.1, taking into account the parameter values

(e,t) = (2, 0) and
(e,u,v) = (X,0,X),(X,0,2), (0, 1, 0), (0, 0, 2), (0, 0, 3).

P 0-l)/2 B np(X,B) 7=i 4<-3 f>x
31
io3 125 84 35 dt = -3
52 io4 1396 1034 + 343
io5 14565 11286 + 3215 + 6
73 106 149204 118455 + 30 559 + 19
IO3 58 52 + 64
IO4 665 617 + 6 190
io5 7099 6 686 + 48
IO6 73252 69365 + 413
io3 31 30 + 3 887
IO4 435 411 + 1
io5 4 709 4 503 + 24
IO6 49 607 47 595 + 206
2012

The result «3(1, 106) = 149204, in comparison to the number 148 709
claimed in [5, Table 5.2, p. 321], shows that 495 complex cubic discriminants
of multiplicity 1 were missing from the original version of [5].

The series of single complex dihedral discriminants starts with -23 for
p = 3, with +2209 = (-47)2 for p = 5, and with -357 911 = (-71)3 for
p = 1. The minimal discriminants of arbitrary quintic fields with signature
(1,2), resp. septic fields with signature (1, 3), are somewhat smaller and not
((P - l)/2)th powers: +1609, resp. -184 607. Unramified extensions are
dominating, 79.5% for p = 3, 94.7% for p = 5, and 95.9% for p = 1, and
this effect even seems to increase with the value of the prime p . The minimal
examples of the rare ramified extensions are dk = -XX with f = 2 for p = 3,
dk = - X5 with / = 5 for p = 5, and dk = -1 with f =1 for p = 1.

(b) By Corollary 3.2,1, np(p - X,B) is the number of (p - l)-tuplets of
ramified cyclic extensions N of degree p over quadratic fields k of p-rank
p = 0 and discriminant \dk\ < B/f2 with two prime divisors c7i, q2 of the
conductor /, such that w < X.

But for the special case p = 3, the number of pure cubic discriminants of
multiplicity 2 must be added. According to Theorem 2.1, we determine this
number by counting primes and products of at most four primes up to certain

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MULTIPLICITIESOF DIHEDRALDISCRIMINANTS 845

bounds, taking into consideration the parameter values

(e,t) = (2,X) and
(e,u,v) = (X, 1, 1), (1, 1, 2), (0, 2,0), (0, 1, 2), (0, 1, 3).

It might be of interest to list explicitly the root discriminants \dL\2l{ß~X)for
the few encountered (p - X)-tuplets of dihedral fields with two prime divisors
of the conductor /, if p = 5, 7 (for p = 3, they start with the well-known

1836 [1]).

There are 20 cases for p = 5 :

105875, 121000, 180500, 287375, 305767, 315375, 315875, 360375,
363000, 496375, 529375, 589875, 605000, 650375, 771375, 786500,
814088, 840500, 841000, 902500,

and 7 cases for p = 1 :

57967, 288463, 576583, 634207, 695604, 985439, 994903.

P (p-l)/2 B np(p -X,B) />1

2 IO3 1 dk <-3 dk = -3
3
IO4 12 0 + 1
io5 167 10 + 2
IO6 1683 157 + 10
IO6 20 1639 + 44
io5 1 20
io6 7
1

7

Here, the authors of [5] announce the value «3(2, IO6) = 1 762 instead of
1683. In fact, the multiplicities of the superfluous 79 discriminants must
necessarily be greater than 2.

(c) The determination of np(p, B) is much more difficult, since it involves
the complete treatment of various cases of p-tuplets of ramified cyclic exten-
sions N of degree p over quadratic fields k of p-rank p > Xand discriminant
\dk\ < B/f2 with conductor / > 1, according to Corollary 3.2. If p = 3, then
further contributions arise from Corollary 3.3. Here, S > X is possible, and
thus the principal pth powers ax, ... , ap of ideals of k must be examined
for each of the relevant quadratic discriminants dk .

(d) By Corollary 3.1, np(p+ X, B) is the number of quadratic fields k with
p-rank p = 2 and discriminant \dk\ < B, i.e., (p + l)-tuplets of unramified
cyclic extensions N\k of degree p with conductor f = X.

In the case of p = 3, where exceptionally not only (p2 - X)/(p - X)= p + X
but also (p - X)2= p + X, we must additionally consider the number of ramified
cyclic extensions N of degree p over quadratic fields k of p-rank p = 0
and discriminant \dk\ < B/f2 with three prime divisors qx,qi, q^ of the
conductor /, such that w < X.

A further additive component for p = 3 is the number of pure cubic discrim-
inants of multiplicity 4, which can again be evaluated with the aid of Theorem
2.1 by counting products of up to five primes below certain bounds, taking into

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846 D. C. MAYER

account the parameter values 2), (0, 3, 0), (0, 2, 2), (0, 2, 3).

(e,t) = (2, 2) and
(e,u,v) = (X,2,X),(X,2,

P (p-l)/2 B np(p+X,B) 7=1 />1 <4= -3
31
104 7 ¿^ < -3 0
52 IO5 216 2
73 IO6 3216 7 + 0 + 11
IO5 39 214 + 0 +
IO6 398 3190 15 +
IO5 1
IO6 97 39
398

1
97

The result «3(4, IO6)= 3 216, in comparison to the number 3 189 given in
[5], reveals the lack of 27 complex cubic discriminants of multiplicity 4 in the
original version of [5].

(e) Finally it should be mentioned, that the authors of [5] announced the
extremely exciting number «3(5, 106) = 7, which caused a lot of confusion.
However, our Example 5 after Theorem 2.1 shows that the minimal occurrence
úf¿= - X6 008 300 of multiplicity 5 (which is only possible for pure and certain
totally real cubic fields) lies considerably outside the range of the computations
in [5].

Bibliography

1. I. O. Angelí,A table of complexcubicfields, Bull.London Math. Soc.5 (1973),37-38.

2. T. Callahan, Dihedral field extensions of order 2p whose class numbers are multiples of p ,
Cañad. J. Math. 28 (1976), 429-439.

3. H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc.
Roy. Soc. London Ser. A 322 (1971), 405-420.

4. R. Dedekind, Über die Anzahl der Idealklassen in reinen kubischen Zahlkörpern, J. Reine
Angew.Math. 121 (1900), 40-123.

5. G. W. Fung and H. C. Williams, On the computation of a table of complex cubic fields with
discriminant D > -IO6 , Math. Comp. 55 (1990), 313-325.

6. F. Gerth III, Cubicfields whoseclass numbers are not divisible by 3 , Illinois J. Math. 20
(1976), 486-493.

7. M. N. Gras, Méthodes et algorithmes pour le calcul numérique du nombre de classes et des
unités des extensions cubiques cycliques de Q , J. Reine Angew. Math. 277 (1975), 89-116.

8. H. Hasse, Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer

Grundlage,Math.Z. 31 (1930),565-582.

9. _, Arithmetische Bestimmung von Grundeinheit und Klassenzahl in zyklischen kubi-
schen und biquadratischenZahlkörpern, Abh. Deutsche Akad. Wiss. Berlin 2 (1950), 3-95.

10. P. Llórente and J. Quer, On totally real cubic fields with discriminant D < IO7, Math.
Comp. 50(1988), 581-594.

11. D. C. Mayer, Table of pure cubic number fields with normalized radicands between 0 and
100000, Univ. Graz, 1989.

12. _, Table of simply real cubic number fields with discriminants between -30000 and
0, Univ. Graz, 1989.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

MULTIPLICITIESOF DIHEDRALDISCRIMINANTS 847

13. D. Shanks, Review of I. O. Angelí, Table of complex cubicfields, Math. Comp. 29 (1975),
661-665.

14. O. Taussky, A remark concerning Hilbert's Theorem 94, J. Reine Angew. Math. 239/240

(1970), 435-438.
15. B. M. Urazbaev, The distribution of the discriminants of cyclicfields of prime degree, Izv.

Akad.Nauk Kazakh.SSR,Ser. Fiz.-Mat.,1 (1969),8-14. (Russian)

Department of Computer Science, University of Manitoba, Winnipeg, Manitoba,
Canada R3T 2N2

E-mail address : [email protected]

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mathematics of computation
volume 58,number 198
april 1992p, ages s55-s58

Supplement to
MULTIPLICITIES OF DIHEDRALDISCRIMINANTS

DANIELC. MAYER

4. Numerical examples

Generally, p = 3 in all examples. The information about the 14 totally real
cubic discriminants with multiplicities 4,6 has been taken from [10]. Data for
the 69 complex cubic discriminants with multiplicities 6,9 has kindly been made
available by the authors of [5] in personal communication.

A common feature of all examples is that the 3-rank of the ring class group
mod/ of the quadratic field k is always p + t + w - 6(f) = 3, whence the sum of
all partial multiplicities equals 1(3?+«+«"-*_ 1) = 13.

Both parts (tu < 1 and w = 2) begin with the unconstrained cases and are
arranged according to increasing quadratic 3-class rank p of k. Correspondingly,
the values of t + w (resp. t) and of u decrease.

For any integer n > 1, denote by On the suborder Z © Xn^(dt + wj of the
maximal order Ok, where u> = y/dk- Then the unit group U(On) is exactly the
intersection Uk l"lQx(n) ■fc*.

In the totally real case, the fundamental unit of fcis denoted by r¡. If p = 0, then
the condition r¡ € On is certainly sufficient for ¿(n) = 0.

Part 1. Applications of Corollary 3.2 to conductors with w < 1.

a) A totally real cubic field with
p = 0, t = 2, u; = 1, «W = 0, u = 3 = t + w,
/ = gi ■Í2 • g3 with ci = 2, q2 - 7, <73= 3.

Here, we have Ik¿{f) = î/jt'^x(/)3- There areno constraints from the fundamental
unit r¡ of fc, since 6(q\ ) = 0, ¿(92) = 0 and ¿(93) = 0. The quadratic discriminant
dk is simultaneously congruent —3(mod9) and congruent 5(mod8), since 2 | /,

(4*-)=2 = -l(mod3).

No, \ d¡, dt 77

1 I8 250228 4677 2 796250463 4 40887672w I 02 n 07 n 03

The partial multiplicities m3(dt,/') of divisors /' of the conductor / can be deter-
mined by means of Corollary 3.2.

/' I 1 gl 92 g3 <Jl92 <?lg3 g2g3 glg2<?3

m3(dk,f) |o I ï Ï 2 2 2^ 4

b) 14 complex cubic fields with

p=l,t + w = 2, Sm¡¡x= 0, u = 2 = í + u>,

/ = gi ■92 with gi = 2.

In this case, /t,3(/)/ix(/)3 = (a). There do not arise restrictions from the single

generating principal ideal cube a of k, i. e., a e Qx(/)-fc? ■k*(f)3, since ¿(gi) = 0

and ¿(92) = 0. All quadratic discriminants dk axe congruent 5(mod8), since 2 | /.

The class numbers of all cubic fields L are divisible by 9.

Typeset by AmS-IfeX

© 1992American MathematicalSociety
0025-5718/92$1:00+ $.25perpage

S55

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