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On the maximal abelian ℓ-extension of a finite algebraic number field with given ramification

Published online by Cambridge University Press:  22 January 2016

Hiroo Miki
Hiroo Miki*
Affiliation:
Department of Mathematics Faculty of Engineering, Yokohama National University
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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. (1) the class number of the cyclotomic Zℓ-extension of k,

  2. (2) the Galois group of the maximal abelian ℓ-extension of k with given ramification,

  3. (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. (4) a certain subgroup Bk(m, S) (cf. § 2) of k×/k×)ℓm for any natural number m (see the main theorem in §3).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 70 , July 1978 , pp. 183 - 202
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1978

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