# Annihilators of the ideal class group of a cyclic extension of an imaginary quadratic field

## TL;DR

This paper investigates elliptic units in cyclic extensions of imaginary quadratic fields, constructing explicit roots to demonstrate annihilation of the p-Sylow subgroup of the ideal class group, advancing understanding of class group structure.

## Contribution

It provides an explicit construction of roots of elliptic units and proves their role in annihilating the p-Sylow subgroup of the class group in such extensions.

## Key findings

- Explicit root of the top generator of elliptic units constructed.
- Annihilation of the p-Sylow subgroup of the class group demonstrated.
- Enhanced understanding of the structure of class groups in cyclic extensions.

## Abstract

The aim of this paper is to study the group of elliptic units of a cyclic extension $L$ of an imaginary quadratic field $K$ such that the degree $[L:K]$ is a power of an odd prime $p$. We construct an explicit root of the usual top generator of this group and we use it to obtain an annihilation result of the $p$-Sylow subgroup of the ideal class group of $L$.