The class group of Dedekind domains

Abstract

A new proof is given of Claborn's theorem, namely that every abelian group is the class group of a Dedekind domain. A variation of the proof shows that the Dedekind domain can be constructed to be a quadratic extension of a principal ideal ring; a Dedekind domain is also constructed that is unrelated in a certain sense to any principal ideal ring. Introduction. Claborn proved in [2] that every abelian group is the class group of some Dedekind domain. In ?1 a new proof is given which is based on a naive geometrical construction. In ?2 the construction is embellished to show that the Dedekind domain can appear as a quadratic extension of a principal ideal ring; a Dedekind domain is also constructed which is unrelated to any principal ideal ring; for precise statements see Theorems 2.1 and 2.3. The last result answers a problem due to Claborn [1]. 1. Every abelian group is a class group. A well-known characterisation of a Dedekind domain as an intersection of valuation rings will be used. A valuation on a field F will mean a normalized exponential valuation; that is to say a homomorphism v of the multiplicative group of F onto the integers under + satisfying v(a + b) > min {va, vb}, with the convention vO = oo. In the notation of Weiss [3] an ordinary arithmeticfield is a field F and a nonempty family {vp I P E A} of valuations of F satisfying (1.1) Va E F, vpa>O for almost all P c A; (1. 2) VP1 0 P2 CA , 3a cF s.t. vpl(a -1) > I; vp2a >I; VP cA, vpa > . Then D = {a E F I VP E A, vpa _ O} is a Dedekind domain. Denoting again by P the set {a I a E D, vpa > O}, P is a prime ideal of D and every nonzero prime ideal of D is of this form; so A freely generates the group of (fractional) ideals of D. The field of fractions of D is clearly F. The class groups of the Dedekind domains we shall construct appear as follows. The generators, or prime ideals, will correspond to certain points in an affine plane, and the relators will correspond to the algebraic curves in that plane; a generator will occur with multiplicity n in the relator corresponding to a given curve if the curve passes through the corresponding point with multiplicity n. We shall therefore need Received by the editors December 8, 1970. AMS 1970 subject classifications. Primary 13D15, 13F05.