Construction of class fields over cyclotomic fields

TL;DR

This paper constructs specific class fields over cyclotomic fields, explicitly determines their degrees, and finds primitive generators using Shimura's reciprocity law and theta constants.

Contribution

It introduces new explicit class fields between ray class fields of cyclotomic fields and constructs their primitive generators.

Findings

Explicit degrees of the constructed class fields are determined.

Primitive generators are constructed using special values of theta constants.

The method employs Shimura's reciprocity law to achieve explicit descriptions.

Abstract

Let $\ell$ and $p$ be odd primes. For a positive integer $\mu$ let $k_\mu$ be the ray class field of $k=\mathbb{Q}(e^{2\pi i/\ell})$ modulo $2p^\mu$. We present certain class fields $K_\mu$ of $k$ such that $k_\mu\leq K_\mu\leq k_{\mu+1}$, and find the degree of $K_\mu/k_\mu$ explicitly. And 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.