On the parity of the class number of an imaginary abelian field of conductor 2~ap~b

摘要

Let p be an odd prime number, and p~(no) the highest power of p dividing 2~(p-1) - 1. for j ≥ 0. Let h*_n be the relative class number of K_n, and h_(n,j) the class number of L_(n,j), respectively. Let n be an integer with n ≥ n_0. We prove that if the ratio h*_n/*_(n-1) is odd, then h_(n,j)/h_(n-1,j) is odd for any j ≥ 0.