P-adic approximation of algebraic integers and residue class rings of rings of integer-valued polynomials

TL;DR

This paper investigates the properties of residue class rings of integer-valued polynomial rings over number fields, characterizing when these rings exhibit the GE2 property through p-adic approximation techniques.

Contribution

It introduces a new characterization of rings of integers where residue class rings of integer-valued polynomials are GE2, using p-adic closure analysis in Galois extensions.

Findings

Residue class rings are GE2 under certain p-adic closure conditions.

Characterization of rings of integers with GE2 property in residue class rings.

Application of Galois extension analysis to integer-valued polynomial rings.

Abstract

Let F:K be a Galois extension of number fields and Q a prime ideal of O_F lying over the prime P of O_K. By analyzing the Q-adic closure of O_K in O_F we characterize those rings of integers O_K for which every residue class ring of Int(O_K) modulo a non-zero prime ideal is GE2 (meaning that every unimodular pair can be trasformed to (1,0) by a series of elementary transformations).