Quotients of Primes in an Algebraic Number Ring

TL;DR

This paper proves that quotients of primes in various algebraic number rings are dense in their respective number sets, extending known results from integers and Gaussian integers to all algebraic number rings.

Contribution

It generalizes the density of prime quotients from integers and Gaussian integers to all imaginary and real quadratic number rings, and further to arbitrary algebraic number rings.

Findings

Quotients of primes are dense in positive real numbers.

Quotients of primes in Gaussian integers are dense in the complex plane.

Density extends to all algebraic number rings.

Abstract

It has been established on many occasions that the set of quotients of prime numbers is dense in the set of positive real numbers. More recently, it has been proved that the set of quotients of primes in the Gaussian integers is dense in the complex plane. In this article, we not only extend this result to any imaginary quadratic number ring, but also prove that the set of quotients of primes in any real quadratic number ring is dense in the set of real numbers. To conclude, we show how to extend these results to an arbitrary algebraic number ring.