Journal of the Mathematical Society of Japan

On the 2-part of the class numbers of cyclotomic fields of prime power conductors


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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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J. Math. Soc. Japan, Volume 64, Number 1 (2012), 317-342.

First available in Project Euclid: 26 January 2012

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Zentralblatt MATH identifier

Primary: 11R18: Cyclotomic extensions 11R23: Iwasawa theory

cyclotomic field Zp-extension parity of class number


ICHIMURA, Humio; NAKAJIMA, Shoichi. On the 2-part of the class numbers of cyclotomic fields of prime power conductors. J. Math. Soc. Japan 64 (2012), no. 1, 317--342. doi:10.2969/jmsj/06410317.

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