Let p be an odd rational
prime and E0= 𝒬() a quadratic imaginary number field. There is a unique Γ−
extension E of E0 for the prime p which is absolutely abelian. For each positive
integer n there is a subfield En of E which is cyclic of degree pn over E0 and by
Iwasawa the exponent of p in the class number of En is of the form μpn+ λn + c for
sufficiently large n. We here examine the analytic formula for the class number of En
and in the case p = 3 give a simple condition implying that ff= 0. It follows easily
from this condition that there are infinitely many imaginary quadratic fields
which have Γ-extensions for the prime 3 with the invariants μ = 0 while
λ ≧ 1.
