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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
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals
Published: Aug 1, 1956
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