Construction of class fields over cyclotomic fields

Abstract

Let $\ell$ and $p$ be odd primes. For a positive integer $\mu$, let $k_{\mu }$ be the ray class field of $k=\mathbb{Q}\left(e^{2\pi i/\ell }\right)$ modulo $2p^{\mu }$. We present certain class fields $K_{\mu }$ of $k$ such that $k_{\mu }\subset K_{\mu }\subset k_{\mu +1}$, and we provide a necessary and sufficient condition for $K_{\mu }=k_{\mu +1}$. We also construct, in the sense of Hilbert, primitive generators of the field $K_{\mu }$ over $k_{\mu }$ by using Shimura’s reciprocity law and special values of theta constants.