proceedings of the american mathematical society Volume 108, Number 3, March 1990 THE RING OF INTEGER-VALUEDPOLYNOMIALS OF A DEDEKIND DOMAIN ROBERT GILMER, WILLIAM HEINZER, DAVID LANTZ AND WILLIAM SMITH (Communicated by Louis J. Ratliff, Jr.) Abstract. Let D be a Dedekind domain and R - Int{D) be the ring of integer-valued polynomials of D . We relate the ideal class groups of D and R . In particular we prove that, if D = 2 is the ring of rational integers, then the ideal class group of R is a free abelian group on a countably infinite basis. If D is an integral domain with field of fractions K, the ring of integer- valued polynomials of D is denoted by Int(D) and is defined to be the subring of K[t] (where t is an indeterminate) consisting of those polynomials /(/) in K[t] such that f(D) ç D. Work on rings of integer-valued polynomials has a long history. In particular, Int(ï), THE ring of integer-valued polynomials, has been studied at least since the work of Ostrowski [O] and Polya [P]. It was well known even then that Int(1) is a free module over Z, with a basis consisting of the binomial polynomials BJt) ,Bx(t), ... , where ß"(i)=C)=i(r_i)""(i_"+i)/"! ' Polya [P] gives a similar result with Z replaced by the ring of algebraic integers in a finite algebraic number field of class number 1. Polya showed that if the integral closure D of Z in a finite algebraic number field is of class number 1, then Int(D) is a free D-module with a basis consist- ing of one polynomial of each nonnegative degree. He called such a Z)-basis for Int(D) a "regular basis". His proof applies for any principal ideal domain D of characteristic zero. Cahen [Ca, §2] proved that if D is a Dedekind domain, Int(D) is a free Z)-module, and that if D is a principal ideal domain, then Int(D) has a regular basis. Received by the editors June 26, 1989. This research was presented to the 845th meeting of the AMS, October 28-29, 1988, Lawrence, Kansas, by Professor Heinzer. 1980 Mathematics Subject Classification (1985 Revision). Primary 13B25, 13F05; Secondary 12B05. Key words and phrases, ring of integer-valued polynomials, invertible ideals, Picard group. The first and second authors gratefully acknowledge the support of the N SF. The third author gratefully acknowledges the hospitality of Purdue University while this work was done. ©1990 American Mathematical Society 0002-9939/90 $1.00+ $.25 per page 673 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use