Internat. J. Math. & Math. Sol.
VOL. 13 NO. 4 (1990) 811-812
811
RESEARCH NOTES
NORMS IN FINITE GALOIS EXTENSIONS OF THE RATIONALS
HANS OPOLKA
Hathematisches Institut
Universitdt Gttingen
Bunsenstrosse 3-5
D-3400 Ottingen
(Received Augusc 22, 1989 and in revised form Becember 12, 1989)
de show that under certain conditions a rational number is a norm
in o given finite Golois extension of the rationals if and only if this number
is a local norm at a certain finite number of places in o certain finite abelion extension of the rationals.
ABST.RACT.
Number fields, norms.
KEY WORDS AND PHRASES.
AH$
CODE. Primary 12AtO, 12A65.
CLA55IFICATION
1980
SUBJECT
INTR ODUCT ION.
Let k be o number field. L. Stern [1] has observed that two finite Golois
extensions L, M of k coincide if and only if the corresponding norm subgroups
coincide. 5o it seems worthwhile to determine the norm
of
L
is certainly a difficult task. We consider the case k
of
which
k
subgroups
1.
LOCAL CONTROL OF GLOBAL NORMS.
Let K/O be a finite Golois extension of degree d and class number h. For
and a given positive integer m we soy
a given finite set of places S of
2 t. n, t |, n odd, if
that the triple (,m,5) is in the special case if m
denotes o
2a 5 and if the cyclotomic extension
2 is not cyclic;
t,
primitive root of unity of order 2
for
and let $ denote the finite set of places of
Let < E
THFOREH.
the
that
Assume
in
K.
ramified
which
is not a local unit and which are
2.
2(21)/
triple
(,d.h,5)
extension
cally in
and h.
2’
ES/
ES/
is not in the special case. Then there is a finite obelion
is o norm losuch that o is o norm in K/ if and only if
is bounded, in terms of d
at oll places in 5. The degree
(Es:)
PROOF. Let H K denote the Hilbert class field of K and let
denote
the maximal central extension of K/ contained in
It follows from
p. 216, Car. III. 2.]3, that ( s o norm in K/ if and only if ,L is o loc01
norm in
at oll places in S. It s well known that o norm subgroup of o
HK/(}.
CK/
CK/O
HANS OPOLKA
812
local extension coincides with the norm subgroup of its maximal aoelian subextension. Therefore we see, [3], p. 9o, (b.9), that there is o finite obelion
at oli places in ) coinsuch that the local extensions of
extension
local extencorresponding
of
the
cide with the maximal obelion subextensions
has the asserted properties.
and such that
sions of
3. A PROBLEM
In connection with the theorem above the following problem arises. For o
given finite Golois extension K/ of degree d and class number h and o given
finite set of places S of Q such that the triple (,d-h,3) is not in the special case, determine the minimal conductor of an obelion extension E/ such
coincide with the maximal
that the local extensions o, F/( at oli places in
field extensions of
class
Hiloert
central
the
local
oOelion suoextensions of
ES/W
ES/Q
ES/
CK/
K/
at all places in o.
ACKNOWLEDGEMENT.
I would like to thank the referee for useful remarks.
REFERENCES
2.
On the norm groups of algebraic number fields, J. of Number
Theory 32 (I 99), 203-219
JAULENT, J.F. L’orithm&tique des l-extensions, Thse Universit& Besonon
3.
NEUKIRCH, J.
I.
STERN, L.
ber dos rinbettungsproblem der olgebaischen Zohlentheorie,
Inv. Math. 21 (1973), 59-116