Note on class number parity of an abelian field of prime conductor, II

Abstract

For a fixed integer n ≥ 1, let p = 2nℓ + 1 be a prime number with an odd prime number ℓ, and let F = F_p,ℓ be the real abelian field of conductor p and degree ℓ. We show that the class number h_F of F is odd when 2 remains prime in the real ℓth cyclotomic field Q(ζ_ℓ)⁺ and ℓ is sufficiently large.