Quotients of primes in an algebraic number ring

Brian D. Sittinger
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 2, Pages 55–62
DOI: 10.7546/nntdm.2018.24.2.55-62
Full paper (PDF, 191 Kb)

Details

Authors and affiliations

Brian D. Sittinger
Department of Mathematics, California State University Channel Islands
1 University Drive, Camarillo, CA 93010, United States

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.

Keywords

• Prime number
• Prime ideal
• Number ring

2010 Mathematics Subject Classification

• 11R04
• 11R44
• 11A41
• 11N05

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