199
Internat. J. Math. & Math. Sci.
(1990) 199-200
VOL. 13 NO.
INVARIANTS OF NUMBER FIELDS RELATED TO CENTRAL EMBEDDING PROBLEMS
H. OPOLKA
Mathematics Inst[tute
Bunsenst rasse 3-5
D-3400 Gott ngen
(Received December 7, 1988)
ABSTRACT.
Every central embedding problem over a number field becomes solvable after
We show that these enlargements can be
enlarging its kernel in a certain way.
arranged in a universal way.
KEY WORDS AND PHRASES.
Central embedding problems,
strict cohomological dimension,
Leopoldt-conjecture.
1980 AMS CLASSIFICATION CODES.
I.
12A55, 12A60
CENTRAL EMBEDDING PROBLEMS.
Let K be a number field and let p be a prime number.
natural number t
Then there is a smallest
t(k,p) depending only on k and p, the so called p-exponent of k,
with the following properties:
E
E(G,Z/p m,c) for the absolute Galois
m
Gal(K/k) is the Galois group of a finite Galois
subextension K/k of k/k which ts ramified only at p and
and where k( )/k is
P
has
i.e.
cyclic,
that E is
is
exponent 2m + t. Recall
there
an
solvable,
m
G / G(c) of G onto the central group extension G(c) defined by the
epimorphtsm
k
k
m
G yields the
co-cycle c:G x G / Z/p such that
composed with the natural map G(c)
G
if and only if the class of (c) becomes trivial in the
given eplmorphism G
k
_,
Brauer group Br(k( )) of k( ),
roots of unity of
k of order
groip of
m
m
m
P
P
P
this means that if
dividing
is an isomorphism then (c) becomes
m
P
trivial under the map
(I)
Every
central
Gal(/k)
group G k
problem
embedding
of k,
where G
pm
2
X m H (G Z/p m)
Xm:Z/pm/
i+nfH2 (Gk z/Pm) r+esH2 (Gk( m )Z!p m)
P
H2(Gk( p
m
k )--Br(k(u
P
m
))
is the map induced by
where X
on cohomology see Hoechsmann, [I]). The exponent
m
of E is the smallest natural number n > m such that the embedding problem E which is
m
n
z/pm+ Z/p n is solvable.
obtained from E by considering the co-cycle c:G x G
Xm
H. OPOLKA
200
In order to prove (I), choose for any natural number
:z/Pm
m
and
the
m
H2 (G,Z/ p
m-m
m such
p
that
Xm.
m
Then we have a map
Br(k(
p
m
and X
relating X
m
diagram
resulting
m an isomorphism
((c)) can
Since
commutes.
)),
be
m
represented by a Galols co-cycle all of whose values are roots of unity, the algebra
class
((c))splits
m
and only if it splits locally at all places above p and
and
m
and
this is the case if it splits at
pm
for all v above p;
E 0 rood
P
(see classels [2], p. 191, 10.5 ff). It is clearly possible to find a smallest integer
(kv
d
P
):kv( m)
((c))
m
d(k,p) depending only on k and p such that
smallest
d(Q,p)
Q we can take d
instance, for k
natural
number n
)
that
m such
solution which is ramified only at p and
the p-exponent of every E
m
is
The p-exponent of E
0 for all p.
the
induced
2m + d.
splits with
is the
has
a
n
intege, r s )0 such that
embedding
The smallest
m
problem E
For
2m + s is called the strong p-exponent of k (if it
exists).
(2)
If p does not divide the class number of
every cyclotomlc field k
Q
(pl)
This can be shown as follows:
t
Let E
m
not
divide
the
the strong p-exponent of
be a central embedding problem for Gk. Then for
t(k,p) the induced embedding problem
that p does
Q(Bp)then
exists and is equal to its (usual) p-exponent.
E2m + t is solvable. The assumption implies
0(Bpl)for every (see lwasawa [3]).
class number of
Therefore the Galols theoretic obstruction to the existence of a solution which is
unramlfled outside p and
as described in Neuklrch [4], (8.1), is trivial.
H2(Gk(P),Q/Z)
The p-adlc Leopoldt conjecture for k implies that
is
the Galols group of
This shows
and
the maximal p-extenslon
which is
Gk(p)
unramlfled outside p
that every central embedding problem Ero for
exponent, (see Opolka [5], (5.2)).
finite?
kP/k
0, where
Gk(p)
has finite p-
Does this imply that the strong p-exponent of k is
If so, how is it related to the usual p-exponent of k?
strong p-exponent of k is finite then
H2(Gk(P),Q/Z)
Conversely, if the
0 and the p-adlc Leopoldt
conjecture holds for k.
REFERENCES
3.
HOECHSMANN, K., Zum Einbettungsproblem, JRAM, 229(1968), 81-106.
CASSELS, J.W.S., A. FrOllch: Algebraic Number Theory, Ac. Press, London, 1967.
IWASAWA, K., A Note on Class Numbers of Algebraic Number Fields, Abh. Math. Sea.
4.
NEUKIRCH,
5.
Inventlones Math., 21 (1973), 59-116.
OPOLKA, H., Cyclotom c Splitting Fields and
Israel J. of Math., 52 (1985), 225-230.
I.
2.
Hamburg, 20 (1956), 257-258.
J.,
Uber
das
Einbettungsproblem
der
Algebralschen
Strict
Zahlentheorle,
Cohomologlcal
Dimension,