On Γ-extensions of algebraic number fields

Let p be a prime number. We call a Galois extension L of a field K a T-extension when its Galois group is topologically isomorphic with the additive group of -adic integers. The purpose of the present paper is to study arithmetic properties of such a T-extension L over a finite algebraic number field K. We consider, namely, the maximal unramified abelian ^-extension M over L and study the structure of the Galois group G(M/L) of the extension M/L. Using the result thus obtained for the group G(M/L)> we then define two invariants l(L/K) and m(L/K) } and show that these invariants can be also determined from a simple formula which gives the exponents of the ^-powers in the class numbers of the intermediate fields of K and L. Thus, giving a relation between the structure of the Galois group of M/L and the class numbers of the subfields of L, our result may be regarded, in a sense, as an analogue, for L, of the well-known theorem in classical class field theory which states that the class number of a finite algebraic number field is equal to the degree of the maximal unramified abelian extension over that field.