  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.

Number Theory