Let $k$ be a real abelian field and $k_{infty}={igcup}_{n{geq}0}k_n$ be its $mathbb{Z}_p$-extension for an odd prime $p$. For each $n{geq}0$, we denote the class number of $k_n$ by $h_n$. The following is a well known theorem: Theorem. Suppose $p$ remains inert in $k$ and the prime ideal of $k$ above $p$ totally ramifies in $k_{infty}$. Then $p{ mid}h_0$ if and only if $p{ mid}h_n$ for all $n{geq}0$. The aim of this paper is to generalize above theorem: Theorem 1. Suppose $H^1(G_n,E_n){simeq}(mathbb{Z}/p^nmathbb{Z})^l$, where $l$ is the number of prime ideals of $k$ above $p$. Then $p{ mid}h_0$ if and only if $p{ mid}h_n$. Theorem 2. Let $k$ be a real quadratic field. Suppose that $H^1(G_1,E_1){simeq}(mathbb{Z}/pmathbb{Z})^l$. Then $p{ mid}h_0$ if and only if $p{ mid}h_n$ for all $n{geq}0$.