Split Primes and Integer-Valued Polynomials

Abstract

Let R be a Dedekind domain with field of fractions K, L = K(α) a finite separable extension of K, and S the integral closure of R in L. Let I be the subring of K[ X] consisting of all polynomials g( x) in K[ X] such that g( R) ⊂ R, and let E α: I → L be the evaluation map defined by E α( g( x)) = g(α). Then E α( I) is precisely the overring of S determined by the prime ideals P of S which are split completely over R and at which α is integral. This answers a question posed by R. Gilmer and W. W. Smith (1985, Houston J. Math. 11, No. 1, 65-74) in connection with the ideal structure of I and solved by them when R = Z and L = Q(√ d).