Ordering of number fields and distribution of class groups: abelian extensions

Weitong Wang

Published 2023-08-22Version 1

When p divides the ordering of Galois group, the distribution of the Sylow p-subgroup of Cl(K) is closely related to the problem of counting fields with certain specifications. Moreover, different orderings of number fields affect the answers of such questions in a nontrivial way. So, in this paper, we set up an invariant of number fields with parameters, and consider field counting problems with specifications while the parameters change as a variable. The case of abelian extensions shows that, roughly speaking, the result of counting abelian fields is a function of the invariant with parameters.