On number fields with $k$-free discriminants

TL;DR

This paper studies number fields with Galois groups and discriminants divisible only by small prime powers, providing new results especially on fields with cubefree discriminants using arithmetic-geometric methods.

Contribution

It generalizes previous work on squarefree discriminants to cubefree discriminants, offering a comprehensive criterion for ramification in Galois cover specializations.

Findings

Characterization of number fields with cubefree discriminants

Effective ramification criterion for Galois cover specializations

Extension of discriminant divisibility analysis to small prime powers

Abstract

Given a finite transitive permutation group $G$, we investigate number fields $F/\mathbb{Q}$ of Galois group $G$ whose discriminant is only divisible by small prime powers. This generalizes previous investigations of number fields with squarefree discriminant. In particular, we obtain a comprehensive result on number fields with cubefree discriminant. Our main tools are arithmetic-geometric, involving in particular an effective criterion on ramification in specializations of Galois covers.