|This paper focuses on number fields and the number rings associated with a particular number field. This topic falls under the study of algebra, and in particular algebraic number theory. The motivation of this paper is to discuss the properties of these number fields in an effort to observe the prime ideal structure. The Dedekind property is most useful in the observation of the prime ideal structure. Along with discussing the structure of these number fields, we will also briefly talk about completions in a number field, which is from the area of topology.