Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

A note on class numbers of algebraic number fields

A note on class numbers of algebraic number fields A note on c1ass numbers of algebraic nwnber Selds By KENKICHI IWAsAwA in Tokyo In the present paper, we shall give a note on c1ass Humbers of algebraic number fields including, in particular, a proof of a theorem of WEBER and its generalization 1). Let k be a finite algebraic number field, K a finite Galois extension of k, and let hand H be class numbers of k and K rcspectively. We first prove the following: I. If there exists a prime divisor P oJ k whichis fully ramified by the extension Klk, then hiH. In particular, Jor any prime lI'umber p, plh ~ plH (or piH ~ pih). 11. IJ, furthermore, Klk is a cyclic extension of p-po'wcr degree and has no ramified prime divisor other than P, then conversely plH ~ plh (or pih ~ pfH). The proof of I is as follows ): Let A and A' be the m:1ximal unramified abelian extensions of k and K respectively. By the class field theory, [A : k] = hand [A' : Kl = H. Since P is fully ramificd by K/k, we have K f\ A = k and the Galois group of KAlk is the direct http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

A note on class numbers of algebraic number fields

Loading next page...
 
/lp/springer-journals/a-note-on-class-numbers-of-algebraic-number-fields-aZWJbdviz4
Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF03374563
Publisher site
See Article on Publisher Site

Abstract

A note on c1ass numbers of algebraic nwnber Selds By KENKICHI IWAsAwA in Tokyo In the present paper, we shall give a note on c1ass Humbers of algebraic number fields including, in particular, a proof of a theorem of WEBER and its generalization 1). Let k be a finite algebraic number field, K a finite Galois extension of k, and let hand H be class numbers of k and K rcspectively. We first prove the following: I. If there exists a prime divisor P oJ k whichis fully ramified by the extension Klk, then hiH. In particular, Jor any prime lI'umber p, plh ~ plH (or piH ~ pih). 11. IJ, furthermore, Klk is a cyclic extension of p-po'wcr degree and has no ramified prime divisor other than P, then conversely plH ~ plh (or pih ~ pfH). The proof of I is as follows ): Let A and A' be the m:1ximal unramified abelian extensions of k and K respectively. By the class field theory, [A : k] = hand [A' : Kl = H. Since P is fully ramificd by K/k, we have K f\ A = k and the Galois group of KAlk is the direct

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Aug 1, 1956

References