ON THE GALOIS MODULE STRUCTURE OVER CM-FIELDS. Jan Brinkhuis. In this paper we make a contribution to the problem of the existence of a normal ...
manuscripta math.
75, 3 3 3 -
347
manuscripta mathematica
(1992)
9 Springer-Verlag 1992
ON THE GALOIS MODULE STRUCTURE OVER CM-FIELDS
Jan Brinkhuis
In this paper we make a contribution to the problem of the existence of a normal integral basis. Our main result is t h a t unramified realizations of a given finite abelian group A as a Calois group Gal(N/K) of an extension N of a given CM-field K are invariant under the involution on the set of all realizations of A over K which is induced by complex conjugation on K and by inversion on A. We give various implications of this result. For example, we show t h a t the tame realizations of a finite abelian group A of odd order over a totally real number field K are completely characterized by ramification and Galois module structure. Introduction By a classical theorem of Hermite each algebraic number field
K
degree
has and
refinement question.
only
finitely
discriminant of
To
this what
many
over
result extent
field
K.
In
one are
extensions
an
attempt
could the
ask
a
following
of
a
Gal(N/K) of
finite abelian group A as a Calois group
given
towards
the
realizations
of
given a tame
field extension N of a given number field K characterized by their
'ramification'
where
the
latter
and
their
is defined to
'Galois be
the
module
structure',
isomorphism class
of
on, the ring of integers in N, as a module over the group ring onA of A over OK? It has been shown in [3] t h a t this question
is
equivalent
to
the 333
following
one,
which
seems
BRINKHUIS
more
restricted
which
are
if
a
is
a
sight:
-at
normal
paper
K
first
unramified
moreover present
at
are
all finite
integral
we o f f e r
primes-
basis
the
CM-field
realizations
over
and K,
following result
(our
definition
of A
of
over
K
which have
rare?
In
the
on this
question
CM-fields
includes
totally real fields).
All unramified integral
basis
realizations
are
of
invariant
,d over K with a normal
under
on the set of all realizations
of A
the
'obvious'
involution
over K which is induced
by complex conjugation on K and by inversion on ,4. In the remainder consequences real,
the
abelian
of
this
result
a
normal
exception
of
exceptions implies
be
of
the
case
restated
as
that
number
integral
basis
over
K,
of
indeed
free
quadratic
exist, ones.
as The
K
with
will
Galois
K
the of
see.
module
totally
unramified
field
extensions
we
is
follows:
real
following
locally
the
totally
an unramified realization of
In
we point out some
a
composita
can
the
result.
can
extensions
never
of this introduction
have
possible K.
Such
This
result
structure
of
of A over K, viewed as an element
class
group
Cg(oK'4)
has
order
either
exp,4 or e x2 p A where exp'4 is the exponent of the group "4. Thus,
in
particular,
odd.
Moreover,
beginning
of
realizations totally
order
returning this
to
is precisely the
question
the
introduction,
exp,4
tame
if
1,41 is
posed
at
the
Galois
algebra
of a finite abelian group ,4 of odd order over a
real
ramification
this
number and
field
Galois
K
are
module
completely structure.
such K and ,4, the unramified realizations
characterized
by
particular,
for
In
of "4 over K have
mutually non-isomorphic Galois module structures. In the other case, nmnely that K is a totally imaginary quadratic
extension
of
a
totally 334
real
number
field,
our
BRINKHUIS
result
has for
Hilbert
if IlK, the
example the following consequence:
class field
of K,
has
a normal integral
basis
over
K, then hK+=l or 2, where hK+ is the class number of K +, the maximal real subfield of K. via
Taking
a
different
Galois
modules
point
of
non-trivial
existence of which cannot
-at
view,
our
elements
result
in
least so f a r -
exhibits
Cs
the
be demonstrated
otherwise. Finally we remark that the old problem of the existence of normal
integral
bases has led in the last twenty years to
an
extensive
literature
of
integers,
with
on
powerful
however
it
seems
that
possible
to
obtain
precise
structure basis
and
in
problem
Galois
methods
this
is
on
the
extensions
the inside
structure
and
on
classical
rings
theorems;
time the
the
of
deep
first
information
particular
of
module
that
it
is
Galois module normal
Hilbert
integral
class
field
of a CM-field.
1.
T h e main r e s u l t Let
rational
QC
of
QC.
embedding
with
complex
requirement totally
number
algebraic
of
Q,
the
field
fields will be considered
Let
a
K
be
CM-field,
an automorphism
that which
is,
of
to
it
be
is
coincides
a for
of K into C, the field of complex numbers, conjugation.
that
This
K is either
imaginary field.
closure
all number
field which has
each
a
an
numbers;
subfields number
be
In
a totally
quadratic the
last
is
extension case
we
CM-field. Let A be a finite abelian group.
335
equivalent real of speak
to
number a
field
totally of
the
a
or real
proper
BRINKHUIS
(1.1)
Definition.
A realization
of
A over
K is a
pair
r=(N,~) consisting of a Galois extension N of K together with an isomorphism ~ from Gal(N/K) to A. Now we choose an embedding of QC into C and we restrict complex conjugation on C to an automorphism
o f QC which we
denote b y c. For each group G and each G-module A we will denote the action of G on A by the left exponential notation
(g,a) -~ ga exponential
(for
geG,
all
notation
more
aeA).
We
generally
to
will
use
the
denote
the
action
left of
the group ring ZG on A. (1.2) K,
D e f i n i t i o n . For each realization r = ( N , ~ ) its
of ,5 over
complex conjugate realization -r=(N,~) is defined as
follows
N= {CnlneN} ~b(w)= r
-1 for all wEGal(N/K)
Warning. Notice the second inverse sign. Inversion
(1.3)
on the group A 'plays the role of complex conjugation'.
Remark. This definition
(1.4)
does
not
depend
on
the
chosen embedding of QC into C.
A realization unramified said
to
at have
all a
r=(N,~)
is
called
unramified
finite
primes.
A
realization
normal
integral
basis
if
if
N/K is
r=(N,~b)
ON, the
ring
is of
integers in N, is a free module on one generator on OKA, the group ring of A over o K.
336
BRINKHUIS
Now we come to the main result of this paper.
(1.5)
Each
Theorem.
unramified
abelian group over a CM-field
realization
of
a finite
with a normal integral basis
is equal to its complex conjugate realization.
This
result
can
also
be
stated
as
follows.
Let
K + be
the
maximal real subfield of K.
Each unramified abelian extension N of a CM-field
(1.6)
K with a normal integral basis over K is Galois over K + and moreover
the
action
of
complex
conjugation
on
Gal(N/K)
by
conjugation in the group Gal(N/K +) is equal to inversion.
In
a
2.9-
previous and
theorem;
in
paper
[4] a
[2] we proved
certain
related a
corollaries
result
much of
is given
weaker
theorem
-theorem
version
(1.5),
of
this
to
be
given
the
following
below, have been obtained already in [2] and [4].
Our
proof
criterion.
of theorem Before
preparations.
We
(1.5)
stating recall
will be it,
we
that
based
on
have
to
Galois
make
theory
some
establishes
a
bijection between the group HI(/2K, A)=Hom(/2K, A), the set of all continuous homomorphisms r from $2K=Cal(QC/K) to .4, and the set of Galois algebras A over K with Galois group A. For each q~eHom(~2K,A ) let Ar be the corresponding Galois algebra and let K s be (Qc)KerS, the fixed field of Kerr of r
One says that
As
is unramified
KS/K
is
at
all
unramified
finite
if the
primes.
the
kernel
field extension
Let
as
be
the
maximal order in As. One says that As has a normal integral basis if a s -~ o/(A as oKA-modules. We consider the action of 337
BRINKHUIS
the group 12K on the ring ?]CA - t h e group ring of A over ?]c the
ring
on
A (c_?]CA) and which acts
action. into
of
The
algebraic
in QC -
the
group
f2K-action
and
which acts
on ?]c (c_TCA) by
inclusion map i from A
?]CA*,
preserves
integers
(with trivial
of
invertible
elements
so
it
a
induces
trivially
the Calois OK-action)
map
in
?]CA,
of
GMois
cohomology groups i•
Hom(J2K, A)=HX(12K,AI) -~ Hx(S2K,TCA*)
Now we can state
the promised criterion (for
the proof we
r e f e r to [4]).
(1.7)
Let
Proposition.
The
r e Horn(OK, A).
following
conditions on r are equivalent (i )
The
Galois
algebra
A~
corresponding
to
r
is
role
in
unramified and has a normal integral basis. (ii) Moreover
i•162 = 1. the
following
result
will
play
a
crucial
our proof. Let G be the product group OK+ x C 2 and let a be the n o n - t r i v i a l
element of Cz. We will consider 12K+ and C2
as subgroups of G. We let the group G act on the group ring ZCA as follows: let 12g act on Z c by the Galois actions, on A trivially,
let a
act on A by inversion and Z c trivially. The
action of the element c ~ = ( c , a ) e G complex
conjugation.
Let
#
be
unity in QC.
338
on ZCA plays the role of the
group
of
all
roots
of
BRINKHUIS
P r o p o s i t i o n . Let ueT?CA * and assume that u m e o K A * f o r
(1.8)
same m > 1. T h e n
1-(~UE #.A To begin with, (x acts on the field components of the
Proof.
semisimple algebra
KA, which are
conjugation. Therefore,
all CM-fields,
as complex
if we project the element 1-~(um) of
KA on one of the field components of KA, then we get an algebraic that
unit
such
l-~(um)
of
units
and
absolute must
value
1; it
roots
of
be
so 1-~u is a
torsion
is a
unity.
standard It
result
follows
that
element in ZCA*. Finally,
it is well-known that the torsion subgroup of ZCA* is #.A. [] *
We write w = c
-1
wc for
all w~S2 K. We define for each
CeHom(~2K, A ) its 'complex conjugate' (~eHom(S2K, A ) as follows -1 for all we/2g.
r162
Now
we
are
ready
to
state
and
prove
the
following
implication which is the key step in the proof.
(1.9)
Proof.
Proposition.
Assume i•162
CeHom(f2K, gl ).
Let
This means that
If
i•162
then
there is an element
u e Z C A * such that
(1.10)
~(w ) = W-lu
Let m be the order of r
V0,)e ~K
Then, by (1.10), u
and so, as (;~cA*)S2K=oKZ1*, 339
m
is fixed by
~2K,
BRINKttUIS
(1.11)
u m e o//A*.
Therefore,
by
proposition
(1.8),
1-C~UE#'A
and
so
we
can
write (:.:2)
1-% = ~p
with ~ e # and p e A. Now we are going to compute for all weE2 K the element
(r
0 =
in
two
different
equal to ~q(~p),
ways.
this
equal last
the
one
hand,
by
(1.12),
0
is
which equals ~,-l~e#. On the other hand, 0
is equal to ~-lu-(~-l)au, (1.10),
On
so, as (w-1)c~=c~(w*-l), this is, by
to r162
expression
as c~ acts is
equal
to
on A by
O(w)r
Comparing
these two outcomes and projecting from # . A = # x A r162
onto A we get
= 1.
This holds for all weE2K, t h a t is, r 1 6 2 By proposition
inversion,
[]
(1.7) and proposition
(1.9) we have
now
p r o v e d the following result. T h e o r e m . Let CeHom(f2r, A ). I f the Galois algebra AV
(1.5)'
corresponding to r
is unramified and has a normal integral
basis, then r = r We have seen
to
be
also just
proved
theorem
theorem
(1.5)' 340
(1.5)
as
with
this
is readily
the
additional
BRINKttUIS
assumption that r is surjective. 2.
Consequences We
start
with
the
implications
of
our
result
normal integral basis problem. These are r a t h e r
for
strong
the if K
is totally real.
(2.1)
Corollary.
extension
of
integral
any
basis
There
totally
with
the
is
real
no
unramified
number
possible
field
exception
with of
abelian a
normal
composita
of
quadratic extensions. Proof. This is just a reformulation of theorem (1.5) for the case that
K is totally
real. To
show this,
it
is convenient
to use the terminology of theorem (1.5)' let eeHom(/2K,A ) be such
that
the
Galois
algebra
AV is
unramified
and
has
a
normal integral basis. To establish the corollary we have to show r
As K is totally real, the automorphism c lies in
~2K and so, as r that
r162162
is ~ = r
~b=~,
Therefore
the
comes down to r
for all ~e~2/~,
conclusion
of
theorem
(1.5)',
This finishes the proof of the
corollary. []
The
possible
exceptions mentioned in (2.1)
can actually
occur for certain K. For example let K be a real quadratic number there
field are
of
odd
precisely
discriminant.
2k - l - 1
It
unramified
is
well-known
quadratic
that
extensions
of K where k is the number of prime numbers dividing the discriminant (see
of
K.
In
proposition (IV 3.5a)
[1] in
we
proved
[6]). 34]
Let
the
S(K)
following be
the
result set
of
BRINKHUIS
number
fields
which
are
composita
of
unramified quadratic
extensions of K. (2.2)
T h e o r e m . Let K be a real quadratic number field of
odd discriminant.
I f the norm of the fundamental unit equals
+1,
is
then
integral If
there
basis
a
unique
over K;
this norm is
-1,
this
field is
N e S( K)
with
a quadratic
then no N e S ( K )
a
normal
extension
of
K.
has a normal integral
basis over K. Now assume that K is a proper CM-field. By class field theory the Galois group over K of HK, the maximal unramified abelian the
extension
class
of
group
K,
of
unramified abelian
is K.
canonically
isomorphic
Therefore,
by
Galois
extensions of K correspond to
to
Cr
theory, subgroups
of CQ~. We recall that the norm map from Cr K to CeK+ is surjective
and
C~K+ to Cr
(2.3)
that
the
the
canonical map
C M - field
K
has
a
normal
integral
basis
the
Let eeHom(~2K, A ) correspond to an unramified Galois
a canonical way over a homomorphism r Translating the
the
then
contains the image of C~K..
algebra As with a normal integral basis. Now r
of r
from
an unramified abelian extension of a
corresponding subgroup of Cr
Proof.
of
has either i or 2 elements.
Corollary. I f
proper
kernel
conclusion of theorem
we get r162 action
of
complex
factorizes in
from CeK to
(1.5)',
r162
A.
in terms
-1 for all xeCQ~, where - denotes conjugation
on
CdK.
Therefore
r
By the remarks preceding the corollary it follows
that Ker r
contains the image of Cdn+. [3 342
BR/NKHUIS
In particular,
letting
hK+ be
the
class
number
of K +,
we get the following result.
(2.4)
If
Corollary.
HK,
the
Hilbert
class
field
of
a
proper C M - f i e l d K, has a normal integral basis over K, then hg+----1 or 2. Proof.
Immediate
from
corollary
(2.3)
and
the
remarks
preceding it. []
For
unramified abelian
extensions N/K,
the
oKA-module
oN - w h e r e AI=Gal(N/K)- is locally free (or, what is the same here,
projective)
class
can
be
of
rank
viewed
as
an
one.
Therefore
element
of
its
the
isomorphism
following finite
abelian group, Cg(oKA), the class group of the ring ogA. It is trivial basis.
precisely
The
if
results
the
above
element is non-trivial,
extension show
that
has in
so one is naturally
a
normal many
integral
cases
led to
this
ask what
its order in the group Cg(OKA ) is. We will denote this order by
ord(o N).
We
can
give
the
following
divisibility
result
for it.
(2.5)
Corollary.
Let K be a CM-field
and A
a finite
abelian group. Let an unramified realization Gal(N/K) .~ A of A over K be given and let r be the corresponding element in Hom(~2n, A). Then ord(0r
[ord(ON) [ordr
Proof. Let H} be the maximal unramified abelian extension of K. We will view Hom(Gal(H~/K),A) as a subgroup of Hom(~2K, A ) via
inflation.
Let g be
the map 343
from ttom(Gal(H~/K),A)
to
BRINKHUIS
Cd(oKA) which sends each r
to the o/r
class of
ere, the maximal order of the Galois algebra As. We have to p r o v e that, for all subjective r (2.6) This
one has
ord(r162 Iordg(r holds
in
fact
more
ordr
generally
without
the
surjectivity
condition: it is known that g is a homomorphism, moreover we have p r o v e d that (2.6)
is an
the condition g ( r
elementary
group
implies r 1 6 2
theoretic
consequence
clearly of
these
facts. [] If
K
is
totally
real,
this
result
amounts
to
the
following one. (2.7)
I f N is an unramified abelian extension
Corollary.
of a totally real number field K, then ord(oN) is k or v2k where k
is
the
exponent
of
the Galois group
of N/K.
In
particular, if [N:K] is odd, then o r d ( o g ) = k . Finally beginning
we
turn
of the
to
introduction
the following definitions. a
finite
abelian
called tame F,
the
if for
to
mentioned
this paper.
In
at
the
[3] we made
F/E of number field is
An extension
each
finite prime
index
is
not
A tame realization
b e a pair (M,r together
question
Let K be a CM-field and let A be
group.
ramification
characteristic.
the
of E which ramifies
divisible
by
the
in
residual
of A over K is defined to
consisting of a tame Galois extension M of K
with an isomorphism from Cal(M/K) to A. Two tame
realizations
(M,r
and
{N,~p)
of
A over
K are
defined
to
h a v e the same ramification if they become isomorphic under a suitable degree,
unramified
base
field
extension
L
of
K
of
in the following sense: the tensor products L| 344
finite and
BRINKHUIS
L|
N are
isomorphic
as L - A - a l g e b r a s ,
that
is, there
isomorphism of L-algebras from LOKM to L| A - a c t i o n s . Two tame realizations (M,r are
said
to
have
the
same
Galois
is an
which preserves
and (N,~b) of A over K module
structure
if
the
maximal orders OM and oN are isomorphic as modules over the group ring o/cA. We proved the following result in [3].
(2.8)
Two tame realizations (M,r
have
the
same
ramification
and
and (N,~p) of A over K the
same
Galois
module
structure if and only if i•162215 ,•
Here ~
is as
in proposition
and ~b with the
(1.7) and
we have
identified
r
elements of Hom(~K,ZI) which one gets from
them b y inflation. The over
K
question, are
module
to
what
characterized
structure
is
extent by
tame
their
therefore
realizations
ramification
essentially
of
and
equivalent
A
Galois to
the
question how far the homomorphism i • is removed from being injective. implies
The
result
r162
gives
of
the
present
information
in
a
paper positive
that
i•162
direction.
For
example, we thus get the following result.
(2.9)
Corollary.
Let
and let Zl be a finite tame
realizations
of
K
be
a
totally
abelian group A
over
K
real
number
of odd order.
are
field
Then the
characterized
by
their
ramification and their Galois module structure.
Proof. Let (M,r K
with
structure.
the
and (N,~P0) be tame realizations of A over
same
Let r
we get from r
ramification
and
the
same
Galois
module
resp. ~p be the element of Hom(S2K, A ) which resp. ~P0 by inflation. 345
Then b y (2.8) we get
BRINKHU/S
i•162 = i•
and
proposition (1.9),
so
i•162 -1) = 1.
r162
by
AS K is totally real, this is
equivalent to the condition (r order, r
Therefore,
that is r
and so, as A has odd
This proves (M,r
[]
In particular we get the following result. (2.10) and
Corollary. A
a
unramified
finite
Let
K
abelian
realizations
be a group
of
A
totally of over
real
odd
number
order.
Then
field the
K
have
mutually
and
non-isomorphic Galois module structures.
REFERENCES
[1]
J. Brinkhuis, Embedding problems doctoral dissertation, Leiden (1981)
Galois
modules,
[2]
J. Brinkhuis, Normal integral bases and conjugation, J. reine angew. Math. 375/376, (1987)
complex 157-166
[3]
J. Brinkhuis, Galois module structure obstruction to a local-global principle, to the Journal of Algebra
[4]
J. Brinkhuis, C M - f i e l d s and (1989)
[5]
L. Childs, The group of unramified Kummer extensions of prime degree, Proc. L.M.S. 35/3, 407-422 (1977)
[6]
A. Fr6hlich, Galois module structure integers, Ergebnisse der Math. 3, 1 (1983)
[7]
V. Fleckinger et T. Nguyen Quang Do, Bases nor'males, unitds et conjecture faible de Leopoldt, preprint (1990)
as the appear in
Unramified abelian extensions of their Galois module structure, preprint
346
of
algebraic
BRINKHUIS
[8]
I.
Kersten
CM-fields,
J.
and J. Michalis Number Theory,
Zp- extensions 32,
no.
2,
of
131-150
(1989)
[9]
Kersten and J. Michaligek: On Vandiver's conjecture and 7p-extensions of Q(~pn), J. Number Theory, 32, no.
I.
3, 371-386 (1989) [10]
L.
Galois
McCulloh,
extensions,
J.
reine
module angew.
structure Math.
of
abelian
375/376,
259-306
(1987)
[11]
M.J.
Taylor,
integers
of
On Fr6hlich's conjecture for rings of tame extensions, Invent. Math. 63, 41-79
(1981)
[12]
M.J.
Taylor,
arithmetic
The Galois module structure of certain principal homogeneous spaces, preprint,
(1990)
Jan Brinkhuis Econometric Institute Erasmus University P.O. Box 1738 3000 DR Rotterdam The Netherlands
(Received February 7, 1991; in revised form January 23, 1992)
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