On Orders in Number Fields: Picard Groups, Ring Class Fields and Applications

Abstract

In this article, we focus on orders in arbitrary number fields, consider their Picard groups and finally obtain ring class fields corresponding to them. The Galois group of the ring class field is isomorphic to the Picard group. As an application, we give criteria of the solvability of the diophantine equation $p=x^2+n{y}^2$ over a class of imaginary quadratic fields where $p$ is a prime element.

13 pages, this version (date: Aug 8, 2014; git commit:20140627a/db806a6) has been accepted for publication in SCIENCE CHINA Mathematics on Jan 4, 2015. arXiv admin note: substantial text overlap with arXiv:1405.7109

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