J
OURNAL DE
T
HÉORIE DES
N
OMBRES DE
B
ORDEAUX
H
ERSHY
K
ISILEVSKY
J
ACK
S
ONN
On the n-torsion subgroup of the Brauer group
of a number field
Journal de Théorie des Nombres de Bordeaux, tome
15, n
o1 (2003),
p. 199-204
<http://www.numdam.org/item?id=JTNB_2003__15_1_199_0>
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
On
the
n-torsion
subgroup
of the Brauer
group
of
anumber
field
par HERSHY KISILEVSKY et JACK SONN
RÉSUMÉ. Pour toute extension
galoisienne
Kde Q
et tout entierpositif
npremier
au nombre de classes deK,
il existe une exten-sion abélienne L de Kd’exposant
n telle que le n-sous-groupe detorsion du groupe de Brauer de K est
égal
au groupe de Brauerrelatif de
L/K.
ABSTRACT. Given a number field K Galois over the rational field
Q,
and apositive integer n prime
to the class number ofK,
there exists an abelian extensionL/K (of
exponentn)
such that then-torsion
subgroup
of the Brauer group of K isequal
to the relative Brauer group ofL/K.
1. Introduction
Let K be a
field,
Br(K)
its Brauer group. IfL/K
is a fieldextension,
then the relative Brauer group
BR(LIK)
is the kernel of the restriction mapTesLjK :
Br(K) - Br (L) .
Relative Brauer groups have been studiedby
Fein and Schacher
(see
e.g.[2,
3,
4].)
Every
subgroup
of Br(K)
is a relativeBrauer group
B r ( L / K )
for some extensionL / K [2],
and thequestion
arisesas to which
subgroups
ofBr(K)
arealgebraic
relative Brauer groups, i. e. ofthe form with
L/K
analgebraic
extension. Forexample
ifL/K
is a finite extension of numberfields,
thenBr(L/K)
is infinite[3],
so nofinite
subgroup
ofBr(K)
is analgebraic
relative Brauer group. In [1] thequestion
was raised as to whether or not the n-torsionsubgroup
of the Brauer groupBr(K)
of a field K is analgebraic
relative Brauer group. Forexample,
if K is a(p-adic)
localfield,
thenBr(K) * Q/Z,
sois an
algebraic
relative Brauer group for all n. This is notsurprising,
since this Brauer group is "small" . Acounterexample
wasgiven
in[1]
for n - 2 and K a formal power series field over a local field. Somewhatsurprisingly,
turned out to be analgebraic
relative Brauer group.Manuscrit reçu le 20 décembre 2001.
The research of the first author was supported in part by grants from NSERC and FCAR. The research of the second author was supported by the Fund for the Promotion of Research at
For number fields
K,
theproblem
is apurely
arithmetic one, because of thefundamental
local-global
description
of the Brauer group of a number field.In
~1~,
it wasproved
that is analgebraic
relative Brauer group forall
squarefree
n. In this paper, we prove thefollowing
affirmative result for number fields:given
any number field K Galoisover Q
and any nprime
to the class number of
K,
Brn(K)
is analgebraic
relative Brauer group,in fact of an abelian extension of K. In
particular,
Brn Q)
is analgebraic
relative Brauer group for all n.
2.
Algebraic
extensions of K with localdegree n everywhere
Theorem 2.1. Let K be a numberfield
Galoisover Q
with class numberttK.
Let be aprime number, relatively prime
tohK
and let r be apositive
integers.
There exists an abelian extensionL/K
of
exponent
£T such thatthe local
degree of
L/K
at everyfinite
prime
equals
î’.If
e =2,
then L canbe taken to be
totall y complexe
Proof.
Let be thecyclotomic
extensionof Q
obtainedby
adjoining
alll-power
roots ofunity
toQ.
Let s be apositive integer.
For lodd,
letkas
be theunique
subfield of ofdegree e$
overQ.
If /!, =2,
there arethree elements of order 2 in one
fixing
the maximal realsubfield,
onefixing
and a thirdfixing
acyclic totally complex
extension
of Q
ofdegree 2s,
which we define to be(As
usual,
pm denotes the mth roots ofunity.)
Then É is theunique prime of Q
ramified inks
and it istotally
ramified.Choose s such that
Lo =
hasdegree .~’’
over K. Then theprimes
of .Kdividing
haveisomorphic completions,
and since e isprime
to
hK, they
are alltotally
ramified inLo/K.
In the case e =2,
the realprimes
are also ramified inLo/K.
Let E be the extension of K obtained
by adjoining
Lo
and theroots of all the units of K
(including
the Ê’th roots ofunity).
Let S’ be the(infinite)
set ofprimes
of K whichsplit completely
in E.For p
E ~ consider the£-ray
class fieldRp
with conductor p, i. e. thel-primary
part
of the ray class field with conductor p. Since the class numberhK
of K isprime
to.~,
p istotally
ramified inFurthermore,
the.2-ray
class groupis
isomorphic
to thee-part
ofI~~ -
(the
multiplicative
group of the residuefield)
modulo theimage
of the unit group of K.By
choice of p ESS’,
the absolute normN(p)
iscongruent
to 1 modulo Ê’ and all unitsare powers in
K*.
HenceJ~
has a(unique cyclic)
quotient
of order£T. We define LP to be the
corresponding cyclic
subextension ofdegrees
ofRp.
Let 1
= ~ be
one of theprime
divisors of Ê in I~. Consider the conditionpower in the ray class group mod p. But since e is
prime
to h =hK,
thisis
equivalent
to theprincipal ideal [h
=(a),
a EK*, being
an ~’’thpower in the ray class group mod p. Since all the units of K are eth powers
modulo p, this is
equivalent
to abeing
an .e’’th power in.Kp,
which for p E ~’ isequivalent
to p
splitting completely
in Denote the acorresponding
to ~
by
ai.Let ~’’ C ,S be the set of
primes
of K thatsplit
completely
inE’ _
E(
lz;al,...,
lytat).
Theprime
divisors 4
of É in .Ksplit completely
in LPif p e SI.
We now define
recursively
asubsequence S’o
ofprimes
ofS’ .
Webegin
with any
prime
pi such that >Af (4-)
four 4
dividing
~.(As
above,
N
denotes the absolutenorm.)
We claim there exists a
prime
P2E S’
withN(~2)
>satisfying:
(a)
P2splits completely
inLP’
(b) pi
splits completely
inLP2;
(c)
q is inert in L~2 for allprimes
q #
pi,Lï
of .~’ with absolute normN(a) _
JV(P,)-To prove the
claim,
we reduce it to anapplication
of Chebotarev’sdensity
theorem.
Arguing
asabove,
(b)
isequivalent
to pibeing
an £Tth power in the ray class group mod p2. But since ~ isprime
to h =hK,
this isequivalent
to the
principal
idealph
=(c),
c EK*,
being
an .eTth power in the ray class group mod P2. Since all the units of K are £Tth powers modulo .p2, this isequivalent
to cbeing
an Ê’th power inK;2’
which isequivalent
to p2splitting completely
inlVê).
Thus(b)
is a Chebotarev conditioncompatible
with(a).
We now consider
(c).
We want all(these
arefinitely
many)
to be inert inLP2 .
For this it suffices that q be inert inMP2,
where MP2 is the subextension of L~2 ofdegree ~
over K.As
above,
ifq~ _ (b), b
eK*,
this means that(b)
is not an lth power inthe ray class group mod p2, i. e. b is not an Êth power in
K;2
(again
sinceall units are £Tth powers in
K;2)’
i.e. P2 isnonsplit
inK(~.c.~, ‘ b).
Sincep2
splits
in this isequivalent
to P2being nonsplit
inFor this Chebotarev condition to be
compatible
with(a)
and(b),
it sufficesthat the fields and
qh
=(b), q =1=
pl,_
be
linearly disjoint
overK(lie,).
Let ql, ... , qu be the
primes
of K distinct from pl,[1,’
...It
of absolute norm less than orequal
to thatof pi,
and letqi -
(bi ) ,
i =1, ... , u.
Set ~’ - We show first that the fields
(K’(fl) : 1
=are
linearly
disjoint
over K’. If not, thenby
Kummertheory
we havean
equation
Ilu b2
a~.
Taking
ideals inK’,
we have =(x)£.
SincehK
isprime
to.~,
and theprimes
qi are unramified inK’,
we see that Ê must divide all the ei, contradiction. SetF2 :=
K’ ( ‘ bl, ... , ‘ bu) .
It remains to show thatFi
~1F~ -
K’.
Ifnot,
then there is a commoncyclic
subextension
F3
g
Fi n F2
with[F3 :
K’] =
é. On the onehand, F3
isof the form with not all ei divisible
by
l. For suchan z,
theprime
divisors of qi inK’
ramify
inF3.
But theonly primes
ramifying
inFI
are divisors
of pi,
It,
contradiction. Thus thedisjointness
assertion isproved.
This shows the existence
Of P2 satisfying
(a),(b),(c).
We now assume
inductively
that n > 2 and Pl,..., p.-j ESI
with i =1, ... ,
n -2,
have been chosen such that(1i)
pisplits completely
in L~j forall j
i,
i =2, ... ,
n - 1(bi)
pj
splits completely
inL’pi
forall j
i,
i . =2, ... , n -
1(ci)
q is inert in for allprimes
qsatisfying
N(Pi-2)
N{q) N(Pi-1),
q ~ p~i, ~ = 2,..., ~ - 1
(take
po =1)
(Note
and(bi )
together
say pisplits
completely
in L’pj for all i~ j,
1
n-1.)
Cl aim: There exists a
prime
pn E
S’satisfying
(an) ,(bn) ,(cn) .
The
argument
is similar to that for P2:(an)
is satisfied if andonly
if Pnsplits completely
in thecomposite
LPn-1. e(bn )
is satisfied if Pnsplits completely
inKsi(
wherep~ = (c,), ~=1,...~-!
(cn )
is satisfiedif pn
remains inert in for eachqh =
(b),
q ~
with In order to
apply
the Chebotarevtheo-rem we need the linear
disjointness
of LPI ... LPn-1 -E’( ‘T cI, ... ,
and the over for all the above b’s. Since the
(ci )’s
and the
(b)’s
are distinctprime
ideals raised to the powerh,
theprevious
argument
goesthrough,
proving
the claim.We therefore have an infinite sequence
So
= ofprimes
of S’
satisfying
(i)
pisplits completely
inLPJ
for all 1~
j,
and(ii)
q is inert in Lpn for allq #
with.J~(q)
Now take L to be the
composite
ofLo
and all the LP-. We check the localdegrees
ofL/K:
For p =
pi ESo,
L contains LPi which istotally
ramified ofdegree
¡;r atp , p
splits
completely
inLa,
andby
(i), ~
splits completely
in L~for j 1= i,
For
p e
So,
p notdividing e, p
is unramified inL,
soLp/Kp
iscyclic
ofexponent
dividing
2T. There exists apositive integer n
such thatN(Pn).
By
(ii),
p is inert in LPn+l hence[Lp : Ke]
= e.For p
dividing ~,
p istotally
ramified inLo
andsplits completely
in all the hence[Lp :
Kip] =
e.Lo
istotally complex,
hence so is L. Thiscompletes
the
proof
of Theorem 2.1. 0Remark 2.2. The
hypothesis
that K is Galoisover Q
guarantees
that allprimes 4
dividing e
in K haveisomorphic completions,
which is all thatis needed in the
proof.
Also we can haveL/K
unramified at the infiniteprimes by
choosing
the maximal real subfield of inplace
ofks.
3. The n-torsion
subgroup
of the Brauer group of K Theorem 3.1. Given a numbersfield
K Galoisover Q
and apositive
in-teger
nprime
to the class numbersof K,
there exists an abelian extensionL/K (of
exponent
n)
such that the n-torsionsubgroup of
the Brauer groupof K
isequal
to the relative Brauer groupof
L/K.
Proof.
Consider the case n =2’’, 2 prime. By
Theorem2.1,
there exists anabelian l-extension
L/K
whose localdegree
at every finiteprime
is£T,
and is 2 at the realprimes
if e = 2. It follows from the fundamental theoremof class field
theory
on the Brauer group of a number field that Lsplits
every
algebra
class of orderdividing
f!,
andconversely,
anyalgebra
classsplit by
L has orderdividing
er. Forgeneral
n, the theorem follows from astraightforward
reduction to theprime
power case(see [1]).
0Remark 3.2. For K =
Q,
Theorem 3.1 says thatBrn(Q)
is analgebraic
relative Brauer group for all n. The
proof
of Theorem 2.1 is more concretein this case because the ray class fields involved are
simply
thedegree 1
subfields of
Q(pp)
with p - 1(mod ~).
Theorem 3.1 wasproved
in[1]
for the case n
squarefree, K
=Q.
The case n = 2 wasproved
thereby
constructing
L /Q
with localdegree
2everywhere
except
perhaps
at theprime
2. We aregrateful
toRomyar
Sharifi and David Ford(independently)
for a construction ofL/Q
with localdegree
2everywhere,
including
2,
theidea of which was instrumental in the
proof
of Theorem 2.1.References
[1] E. ALJADEFF, J. SONN, Relative Brauer groups and m-torsion. Proc. Amer. Math. Soc. 130
(2002), 1333-1337.
[2] B. FEIN, M. SCHACHER, Relative Brauer groups I. J. Reine Angew. Math. 321 (1981), 179-194.
[3] B. FEIN, W. KANTOR, M. SCHACHER, Relative Brauer groups II. J. Reine Angew. Math. 328
[4] B. FEIN, M. SCHACHER, Relative Brauer groups III. J. Reine Angew. Math. 335 (1982), 37-39. Hershy KISILEVSKY Department of Mathematics Concordia University 11455 de Maisonneuve Blvd. West Montreal Canada kisilevomathstat.concordia.ca Jack SONN Department of Mathematics Technion 32000 Haifa Israel E-mail : sonnCmath.technion.ac.il