arXiv: Number Theory | 2019
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 $q \\equiv 1 \\pmod \\ell$ 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 $H$.