Journal of the Mathematical Society of Japan
Online ISSN : 1881-1167
Print ISSN : 0025-5645
ISSN-L : 0025-5645
On the 2-part of the class numbers of cyclotomic fields of prime power conductors
Humio IchimuraShoichi Nakajima
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2012 年 64 巻 1 号 p. 317-342

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Let p be an odd prime number and ℓ a prime number with ℓ ≠ p. Let Kn = Qpn+1) be the pn+1-st cyclotomic field. Let hn and hn- be the class number and the relative class number of Kn, respectively. When ℓ = 2, we give an explicit bound mp depending on p such that the ratio hn-/hn-1- is odd for all n > mp. When ℓ ≥ 3, we also give a corresponding result on the ℓ-part of the relative class number of Kn+). As an application, we show that when p ≤ 509, the ratio hn/h0 is odd for all n ≥ 1.
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© 2012 The Mathematical Society of Japan
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