Primes of Higher Degree

2023/10/08 by Pandey, Prem Prakash, Ram, Mahesh Kumar
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2310.04974

Abstract

Let K/\Q be a cyclic extension of number fields with Galois group G. We study the ideal classes of primes \mathfrakp 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 \mathfrakp 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 ℓ-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.

Related

• Primes of higher degree and Annihilators of Class groups
• Distribution of prime ideals of higher residue degree across ideal\n classes in the class groups
• Lower bounds on the ℓ-rank of ideal class groups
• l-Class groups of cyclic extensions of prime degree l
• The finitude of tamely ramified pro-p extensions of number fields with cyclic p-class groups