Generators for abelian extensions of number fields

TL;DR

The paper constructs universal primitive generators for abelian extensions of number fields, ensuring their traces generate intermediate fields, and introduces methods for finding normal elements, with applications to ray class fields.

Contribution

It introduces a universal primitive generator construction for abelian extensions and a new method for finding normal elements, advancing explicit generator theory.

Findings

Universal primitive generators can be constructed with trace properties

Applications to towers of ray class fields over imaginary quadratic fields

A new method for finding normal elements in abelian extensions

Abstract

Let $U/L$ be a finite abelian extension of number fields. We first construct a universal primitive generator of $U$ over $L$ whose relative trace to any intermediate field $F$ becomes a generator of $F$ over $L$, too. We also develop a similar argument in terms of norm. As its examples we investigate towers of ray class fields over imaginary quadratic fields. And, we further present a new method of finding a normal element for the extension $U/L$.