RELATIONS OF IDEALS OF CERTAIN REAL ABELIAN FIELDS

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 $\prod_{{{\chi}{\in}\hat{\Delta}_k},{\chi}{\neq}1}B_{1,{\chi}{\omega}^{-1}$, 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 $p{\mid}{\prod}_{{\chi}{\in}\hat{\Delta}_{k,p},{\chi}{\neq}1}B_{1,{\chi}{\omega}}-1$, then $A_n{\neq}\{0\}$ 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 $p$.