On number fields with power-free discriminant

Abstract

Given a finite transitive permutation group G, we investigate number fields F/DOUBLE-STRUCK CAPITAL 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.