On Ring Class Fields of Number Rings

Abstract

For a number field K, we extend the notion of the ring class field of an order in K [C. Lv and Y. Deng, SciChina. Math., 2015] to that of an arbitrary number ring in K. We give both ideal-theoretic and idele-theoretic description of this number ring class field, and characterize it as a subfield of the ring class field of some order. As an application, we use it to give a criterion of the solvability of a higher degree norm form equation over a number ring and finally describe algorithms to compute this field.

Entanglement in the family of division fields of elliptic curves with complex multiplication

“…This work has been reformulated using an adelic language by Stevenhagen in[34, § 4]. Finally, the ring class elds H O have been studied for general number elds F by Lv and Deng in[20] and by Yi and Lv in[38].Remark A.3. It is clear from the de nition that for every pair of ideals I ⊆ ⊆ O we have that U I,O ⊆ U ,O , which implies that H I ,O ⊇ H ,O .…”