Let p be an odd prime number, and \({p^{n_0}}\) the highest power of p dividing 2p−1 − 1. Let \({K_n={\bf Q}(\zeta_{p^{n+1}})}\) and \({L_{n,j}=K_n^+(\zeta_{2^{j+2}})}\) 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 \({h_n^*/h_{n-1}^*}\) is odd, then h n,j/h n−1,j is odd for any j ≥ 0.