Let $k$ be a real abelian field and $k_{infty}$ be its $mathbb{Z}_p$-extension for an odd prime $p$. Let $A_n$ be the Sylow $p$-subgroup of the ideal class group of $k_n$, the $nth$ layer of the $mathbb{Z}_p$-extension. By using the main conjecture of Iwasawa theory, we have the following: If $p$ does not divide $prod_{{{chi}{in}hat{Delta}_k},{chi}{ eq}1}B_{1,{chi}{omega}^{-1}$, then $A_n$ = {0} for all $n{geq}0$, where ${Delta}_k=Gal(k/mathbb{Q})$ and ${omega}$ is the Teichm$ddot{u}$ller character for $p$. The converse of this statement does not hold in general. However, we have the following when $k$ is of prime conductor $q$: Let $q$ be an odd prime different from $p$. and let $k$ be a real subfield of $mathbb{Q}({zeta}_q)$. If $p{mid}{prod}_{{chi}{in}hat{Delta}_{k,p},{chi}{ eq}1}B_{1,{chi}{omega}}-1$, then $A_n{ eq}{0}$ for all $n{geq}1$, where ${Delta}_{k,p}$ is the $Gal(k_{(p)}/mathbb{Q})$ and $k_{(p)}$ is the decomposition field of $k$ for $p$.