On class groups of abelian extensions

Abstract Let K be an absolutely abelian number field, and t be a positive integer relatively prime to $[K: \mathbb {Q}]$ . We provide a method to use the t -rank of the class group of K to obtain a lower bound on the t -rank of class groups of any cyclotomic fields (or real cyclotomic fields) that contain this K . As an application, for any odd integer $t> 1$ , we provide a family of cyclotomic fields whose class groups have t -rank at least $2$ . Furthermore, we prove that the class number of $\mathbb {Q}(\zeta _{p^e} + \zeta _{p^e}^{-1})$ cannot be equal to $2$ , where p is a prime, $e \geq 1$ is an integer, and $\zeta _{p^e}$ denotes a primitive $p^e$ -th root of unity. We also provide an upper bound for the class number of a real cyclotomic field.