On the maximal abelianℓ-extension of a finite algebraic number field with given ramification
Hiroo Miki · Nagoya Mathematical Journal · 1978
Letkbe a finite algebraic number field and letℓbe 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 cyclotomicZℓ-extension ofk, (2) the Galois group of the maximal abelianℓ-extension ofkwith given ramification, (3) the number of independent cyclic extensions ofkof degreeℓ, which can be extended to finite cyclic extensions ofkof anyℓ-power degree, and (4) a certain subgroupBk(m, S) (cf. § 2) ofk×/k×)ℓmfor any natural numberm(see the main theorem in §3).