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Introduction to Iwasawa theory
by Keiichi Komatsu
2
. Then Q (  m ) is a cyclic extension
2n  2
of
whose degree over
is 2 n . Since  m2 1   m2  2 , we have ( m1 )  ( m ) .
Weber proved the following theorem (maybe in 1900):
Let n be a non-negative integer and  n  2 cos
Theorem 1. The class number of (  m ) is odd.
In our lectures, we prove the above theorem not using class field theory.
After 50 years, Iwasawa proved the following theorem using class field theory:
Theorem 2 [2]. Let k be an algebraic number field, K a finite Galois extension of k and
the Galois group G ( K / k ) is a p-group (p: any prime). Assume there is at most one prime,
which ramifies in K over k. If p does not divide the class number of k, then p does not
divide the class number of K.
We give a proof of Theorem 2 in our lectures. We note that Theorem 2 implies Theorem
1.

( m ) . The Galois group G (
(2)

integer ring 2 .
Weber was the first mathematician who encountered
p
Now, we put
(2)


/ ) is isomorphic to the 2-adic
n 0
-extension.
Now, we consider a tower of number fields
k  k0  k1   kn 
such that km is a cyclic extension of k of degree pn.
We put k( p ) 

km . We call k( p ) / k a
p
-extension of k. Main purpose of our lectures
m 0
is to prove the following theorem:
Theorem 3. Let Am be the p-sylow subgroup of the ideal class group of km . Then there
exists integers  、  and  such that the order of Am is pm  p
n.
m

for sufficiently large
In the rest of our lectures, we may discuss p-adic L-function, Iwasawa main conjecture
and Iwasawa theory of CM-elliptic curves ( [1] and [3] ).
Reference:
[1] Coates, J. and Wiles, A., On the conjecture of Birch and Swinnerton-Dyer. Invent.
Math. , 39(1977), 223-251.
[2] Iwasawa, K., A note on class numbers of algebraic number fields. Abh. Math. Sem.
Univ. Hamburg, 20 (1956), 257-258.
[3] Washington, L. C., Introduction to cyclotomic fields, 2nd edition, G.T.M., 83,
Springer-Verlag, New York, Heidelberg, Berlin (1997)