Primary rank of the class group of real cyclotomic fields

TL;DR

This paper establishes bounds on the b1b5;-rank of the class group of real cyclotomic fields, extending previous results to a broader setting and relating it to real quadratic subfields.

Contribution

It provides new bounds on the b1b5;-rank of class groups of real cyclotomic fields, generalizing prior work for b5=3 and connecting it to real quadratic subfields.

Findings

Bound on b1b5;-rank in terms of quadratic subfields

Extension of Agathocleous's b5=3 case to general odd primes

Relation between b1b5;-ranks of subfields

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 $H$.