On orders in number fields: Picard groups, ring class fields and applications

Abstract

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 + ny 2 over a class of imaginary quadratic fields where p is a prime element. IntroductionOrders in number fields are widely used in many problems. A typical instance can be found in the book [6] by David A. Cox, where the author was dealing with the solvability of the diophantine equation p = x 2 + ny 2 over Z for an odd prime p and a positive integer n. Using the order Z[ √ −n] in imaginary quadratic field Q( √ −n) along with the corresponding ring class field, Cox gave a beautiful criterion for the equation before, available for arbitrary n > 0.There are many other studies of ring class fields, say [4,3]. However, it seems that the result that there's an isomorphism between the Picard group of an order and the Galois group of it's corresponding ring class field only restricts to orders in imaginary quadratic fields, more generally, CM-fields (a totally imaginary quadratic extension of a totally real field). Here we consider orders in arbitrary number fields and their ring class fields, generalizing methods in [6]. By construction isomorphisms, we identify the Picard group as an generalized ideal class group, and therefore using the class field theory in the ideal-theoretic version we obtain a class field whose Galois group is isomorphic to the Picard group. Definitions and basic facts about orders are given in Section 2, and isomorphisms between groups are studied in Section 3 subsequently. Then we can define the ring class fields in Section 4. The last Section is dedicated to an application, considering the solvability of the diophantine equation p = x 2 + ny 2 over a class of imaginary quadratic fields where p is a prime element. Definitions and Basic FactsWe adopt the standard notations without additional explanation. If R is a ring, we denote as R × the unite group of R. Let K be a number field, o K the ring of integers of K, o an order in K. Denote the group of fractional ideals of K by J K , the principal fractional ideals by P K and the ideal class group by Cl K . For the invertible and principal fractional ideals of o we use J(o) and Definition 2.1. A modulus in K is a formal product m = p p np over all primes p, finite or infinite, of K, where all integers n p ≥ 0 are nonzero only at finite many primes, zero at all complex primes and less than one at all real primes. We may write m = m 0 m ∞ where m 0 is an o K -ideal and m ∞ is a product of distinct real primes of K. If all n p = 0 we set m = 1. The normalization of o is defined as the integral closure of o in K. In our case it is o K , and hence it is finitely generated over o. We call the biggest ideal f of o K contained in o, in other words, × are finite and one haswhere h K = #Cl K is the class number of K. The Connectio...

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“…Let be a number field and an order of of conductor . We construct a class field similar to the ring class field of studied in , except that our congruence subgroup also incorporates the condition that . We consider the modulus of , where is the product of all real places of .…”

“…Hence, . (2): this is [, Proposition 3.4]. (3): the isomorphism from (2) maps to .…”

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