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

Abstract

Let p be an odd prime number and ℓ a prime number with ℓ ≠ p. Let K_n = Q(ζ_pⁿ⁺¹) be the pⁿ⁺¹-st cyclotomic field. Let h_n and h_n^- be the class number and the relative class number of K_n, respectively. When ℓ = 2, we give an explicit bound m_p depending on p such that the ratio h_n^-/h_n-1^- is odd for all n > m_p. When ℓ ≥ 3, we also give a corresponding result on the ℓ-part of the relative class number of K_n⁺(ζ_ℓ). As an application, we show that when p ≤ 509, the ratio h_n/h₀ is odd for all n ≥ 1.