Primary rank of the class group of real cyclotomic fields

Abstract

Let $H=\mathbb{Q}\left({\zeta }_n+{{\zeta }_n}^{-1}\right)$ and $\ell$ be an odd prime such that $q\equiv 1\left(mod\ell \right)$ for a prime factor $q$ of $n$. We get a bound on the $\ell$-rank of the class group of $H$ in terms of the $\ell$-rank of the class group of a real quadratic subfield contained in $H$. At the end we look into few numerical examples.