Primary rank of the class group of real cyclotomic fields

Let $H= \mathbb{Q}(ζ_{n} + {ζ_{n}}^{-1})$ and $\ell$ be an odd prime such that $q \equiv 1 \pmod \ell$ 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 $H$.

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