Analogues of the prime number theorem in certain rings of integers

Abstract

This thesis is dedicated to constructing analogues of the prime number theorem in the rings of integers with negative radicand which are Euclidean domains. First, a brief history of the prime number theorem is given, including a known elementary proof of the prime number theorem and a proof of the prime number theorem for primes in arithmetic progressions. Then, well-known results from commutative ring theory and number theory are given. The author then presents characterizations for prime elements in the relevant rings of integers. The thesis culminates in analogues of the prime number theorem for each respective ring of integers.