Primary rank of the class group of real cyclotomic fields

Abstract

Let $H=\mathbb{Q}({\zeta }_n+{{\zeta }_n}^{-1})$ and $\ell$ be an odd prime such that $q\equiv 1(\mathrm{mod}\ell )$ for some prime factor $q$ of $n$. We get a bound on the $\ell$-rank of the class group of $H$(under some conditions) in terms of the $\ell$-rank of the class group of real quadratic subfield contained in $H$. This is an extension of a recent work of E. Agathocleous (with alternate hypothesis) where she handles $\ell =3$ case. As an application of our main result we relate the $\ell$-rank of real quadratic subfields of $H$.

7 pages, Any comments will be appreciated