Regarding non-unique factorization of integer-valued polynomials over a discrete valuation domain (R,M) with finite residue field, it is known that there exist absolutely irreducible elements, is, elements all whose powers factor uniquely, and non-absolutely elements. We completely constructively characterize the among split polynomials. They correspond bijectively to sets, which we call balanc...

We show that every Dedekind domain $R$ lying between the polynomial rings $\mathbb Z[X]$ and Q[X]$ with property its residue fields of prime characteristic are finite is equal to a generalized ring integer-valued polynomials, is, for each $p\in\mathbb Z$ there exists subset $E_p$ transcendental elements over Q$ in absolute integral closure $\overline{\mathbb Z_p}$ $p$-adic integers such $R=\{f\...

Proof. We will calculate the integral closure of A inside K. We begin with a more general discussion. Suppose C is an irreducible, affine curve, that is, C is some irreducible affine variety of dimension 1. Let A be the ring of regular functions on C, so that A is a noetherian domain of Krull dimension 1. Localization preserves the noetherian condition, as well as the domain condition. Thus, th...

A Dedekind category (locally complete division allegory) D is a category satisfying the following:

In this paper we will study the number-theoretical properties of the expression v1 nkal rcka,, d(p; a I . . . . . an) = ( 1) n/2 ~ cot cot (1) k=l P P and of related finite trigonometric sums. In Eq. (I), p is a positive integer, a~ . . . . . a, are integers prime to p, and n is even (for n odd the sum is clearly equal to zero). There are two reasons for being interested in sums of this type. F...