manuscripta math. 40, 155- 203 (1982) manuscripta mathematica 9 Springer-Verlag1982 INTEGER VALUED POLYNOMIALS OVER A NUMBER FIELD H. Zantema A number field is called a Pdlya field if the module of integer valued polynomials over that field is generated by (fi)i=0 over the ring of integers, with deg(f i) = i, i = 0, i, 2,.. . In this paper bounds on the class numbers and on the number of ramified primes in Pdlya fields are derived. I. INTRODUCTION Let K be a number field and let 0 = 0(K) be its ring of integers. Define t R(K) = { f s K[X3 1 f[0] c 0 }. In section 2 we shall see that R(K) is a free 0-module for each number field K. In this paper we are interested in those fields K for which R(K) has an 0-basis (fi)i=0 with the property that deg(f.) = i, i = 0, I, 2,.. . Such fields we call P61ya fields, l after G. Pdlya, who obtained the first results in this subject in 1919, see [P]. He remarked that a number field is a Pdlya field if and only if for each positive integer i the fractional ideal a consisting of the i-th degree coefficients of elements of --l degree ~ i of R(K), is principal. Hence if the class number h(K) of K is one then K is a Pdlya field. One of Pdlya's main conclusions was that a quadratic field is a P61ya field if and only if all ramified prime ideals are principal. In a paper immediately following Pdlya's paper, A. Ostrowski ([03) proves that K is a P61ya field if and only if the ideal b(q) = n N (s =q is principal for all prime powers q, where p ranges over the prime ideals of 0 and N denotes the absolute ideal norm. If K/~ is OO25-2611/82/OO40/O155/$O9.80 155