| This dissertation focuses on the construction of Cohen-Kaplansky domains. The main result is the creation of several new classes of CK-domains, constructed through the use of power series extensions and localizations. First, we show that rings of the form A&sqbl0;&sqbl0;&cubl0;xai&cubr0; ni=1&sqbr0;&sqbr0; are necessarily CK-domains, provided A is a finite field, and gcd(a1, ..., an) = 1. We illustrate the unique relationship between rings of this form with the Frobenius number of the set {a 1, ..., an}. In the special case of F2&sqbl0;&sqbl0;x2 ,x2n+1&sqbr0;&sqbr0; , we give a complete characterization of the irreducible elements and compute the size of the CK-domain. In the more general case, algorithmic results are obtained for the general F2 case, while partial results are obtained for the arbitrary finite field case. Second, we show that localizations of certain orders of rings of integers are necessarily CK-domains, then we prove there exists a closed form formula for the number of irreducible elements in several different cases of these types of rings. This construction will then allow us to answer a question posed by Cohen and Kaplansky regarding the construction a CK- n domain for every positive integer n. |
