Title: On the maximal abelian<i>ℓ</i>-extension of a finite algebraic number field with given ramification
Abstract: Let k be a finite algebraic number field and let ℓ be a fixed odd prime number. In this paper, we shall prove the equivalence of certain rather strong conditions on the following four things (1) ~ (4), respectively : (1) the class number of the cyclotomic Z ℓ -extension of k , (2) the Galois group of the maximal abelian ℓ -extension of k with given ramification, (3) the number of independent cyclic extensions of k of degree ℓ , which can be extended to finite cyclic extensions of k of any ℓ -power degree, and (4) a certain subgroup B k ( m, S ) (cf. § 2) of k × / k × ) ℓm for any natural number m (see the main theorem in §3).