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62
J. W. S. CASSELS
Now suppose that a E R.H.S.
and that
,!I=$b,,w,ER.H.S.
(b,Ek,).
1
(12.2)
Then for any m, 1 I
m I
N we have
I.@,
,...,
w,,~,P,w,,+~
,...,
d=b:W’,
,...,
R>,
and so
dbi E o,
(1 I
m 5 N)
where
d=D(o,,.
..,w,)Ek.
But (Appendix
B) we have d # 0, and so ldlv = 1 for almost
all u.
For
almost
all u the condition
(12.2) thus implies
b, E o,
(15
m 5 N),
i.e.
R.H.S.
c L.H.S.
This proves the lemma.
[COROLLARY.
Almost all v are unramijied
in the extension K/k.
For by the results of Chapter I a necessary
and sutlicient condition for v to be un
ramified
is that there are yl,. . ., yN E R.H.S. with ID(y,,. . ., yN)lv = 1. And for almost
all v we can put yn = ~f”~.]
13. Restricted
Topological
Product
We describe here a topological
tool which will be needed later:
DEFINITION. Let sZA
(A E A) be a family of topological spaces and for almost
alIt L let On c Q, be an open subset of a,. Consider the space n whose points
are sets u =
h&h
where aA E a, for every A and cr, E On for almost all 1.
We
give Q a topology by taking as a basis of open sets the sets
lm
where rl
c G12,
is open for all rZ and rA = O1 for almost all 1.
With this
topology R is the restricted topologicalproduct of the R, with respect to the O1.
COROLLARY. Let S be a finite subset of A and let Rs be the set of a E fi
with aA E O1 (2 q! S), i.e.
(13.1)
Then G!, is open in Sz and the topology induced in as as a subset of R is the
same as the product topology.
Beweis. Klar.
The
restricted
topological
product
depends
on the totality
of the
0,
but not on the individual
0,:
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View Full Document GLOBAL
FIELDS
63
LEMMA.
Let
0;
c Jz, be open sets defined for almost
all
II and suppose
that OA = 0: for
almost
all 2.
Then the restricted product
of the f2, with
respect to the 0; is the same ast the restrictedproduct
with respect to the 0,.
Beweis.
Klar.
LEMMA.
Suppose that
the RA are locally
compact
and that
the 0,
are
compact.
Then R is locally
compact.
Proof.
The
as
are locally
compact
by (13.1) since S is finite.
Since
n = u C& and the S& are open in 0, the result follows.
DEFINITION.
Suppose
that
measures
,uI
are
defined
on
the
RA with
~~(0,)
= 1 when OA is defined.
We define the product
measure p on R to
be that for which a basis of measurable sets is the
where MA c Sz, has finite
PAmeasure and MA = On for
aImost
all
1 and
where
COROLLARY.
The restriction
of ,u to Q, is just the ordinary product
measure.
14. Adele Ring
(or Ring of Valuation
Vectors)
Let k be a global
field.
For each normalized
valuation
1 1” of k denote by
k, the completion
of k.
If 1 1” is nonarchimedean
denote by D, the ring of
integers of k,.
The adele ring
V, of k is the topological
ring whose under
lying
topological
space is the restricted
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This note was uploaded on 10/26/2011 for the course MATH 8410 taught by Professor Staff during the Fall '11 term at University of Georgia Athens.
 Fall '11
 Staff
 Number Theory

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