Let A be a Dedekind domain with finite residue fields, K it's quotient field, L a finite separable extension of K, and B the integral closure of A in L. The rings of integer-valued polynomials on A and B are known to be Pru fer domains and will be denoted by Int(A) and Int(B), respectively. We will present some connections between the localizations of each of these rings (Theorem 2 and Corollary 4) and in particular we answer the question when Int(A) is a subset of Int(B).
1998 Academic Press
Let A be a Dedekind domain with finite residue fields and K the quotient field of A. We denote by Int(A) the subring of K[X] consisting of all polynomials f (X) such that f (A)/A. Serious study of the ring Int(A) began with papers of Polya [7] and Ostrowski [6] published in 1919. The main focus of these papers is the A-module structure of Int(A) where A is the ring of algebraic integers in a finite extension of the field of rational numbers.
Let L be a finite separable extension of K and B the integral closure of A in L. The rings Int(A) and Int(B) have no obvious connections, Int(A) is not even a subring of Int(B) [5]. In this note we put in evidence some connections between prime ideals of Int(A) and Int(B). We exhibit a relation between these prime ideals and the prime ideals of A which split completely in L, using the essential valuations of Int(A) and Int(B).
If P is a non-zero prime ideal of B and p=P & A, we denote, as usual, e(PÂ p), f (PÂp) the ramification index and the residue degree, respectively, of P over p. The prime ideal p of A splits completely in L if e(PÂ p) f (PÂ p) =1 for all prime ideals P of B such that P & A= p. A valuation w on L is called an extension of a valuation v of K if w(x)=v(x) for all x # K. If m and M are the maximal ideals of v and w, respectively, we denote by e(wÂv) the ramification index e(MÂm) and by f(wÂv) the residue degree f (MÂm). For every valuation v we denote by 1 v the value group and by k v the residue field of v. If A is a Dedekind domain then Int(A) is a Pru fer domain [2] (a domain is called a Pru fer domain if all the localizations with respect to Article No. NT982267