On Ring Class Fields of Number Rings

TL;DR

This paper generalizes the concept of ring class fields from orders to arbitrary number rings in a number field, providing theoretical descriptions, solvability criteria for norm form equations, and algorithms for computation.

Contribution

It extends the notion of ring class fields to all number rings, offering ideal and idele descriptions, and develops methods for solving norm form equations and computing these fields.

Findings

Characterization of number ring class fields as subfields of order-based class fields

A criterion for solvability of higher degree norm form equations over number rings

Algorithms for computing the generalized ring class fields

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.