Primes of Higher Degree

TL;DR

This paper investigates primes of higher residue degree in cyclic number field extensions, revealing their role in generating class groups and providing insights into class number factors, torsion, and structure.

Contribution

It introduces the study of primes with residue degree greater than one in cyclic extensions and shows their significance in understanding class group properties and constructing annihilators.

Findings

Existence of many fields where primes of a fixed higher residue degree generate the class group.

Results on class group rank, factors of class number, and structure based on these primes.

Construction of annihilators of class groups using primes of higher residue degree.

Abstract

Let $K/\Q$ be a cyclic extension of number fields with Galois group $G$. We study the ideal classes of primes $\mathfrak{p}$ of $K$ of residue degree bigger than one in the class group of $K$. In particular, we explore such extensions $K/\Q$ for which there exist an integer $f>1$ such that the ideal classes of primes $\mathfrak{p}$ of $K$ of residue degree $f$ generate the full class group of $K$. It is shown that there are many such fields. These results are used to obtain information on class group of $K$; like rank of $\ell-$torsion of the class group, factors of class number, fields with class group of certain exponents, and even structure of class group in some cases. Moreover, such $f$ can be used to construct annihilators of the class groups.