Abstract Algebra Theory and Applications - Computer Science ...

16Integral DomainsOne of the most important rings we study is the ring of integers. It was ourfirst example of an algebraic structure: the first polynomial ring that weexamined was Z[x]. We also know that the integers sit naturally inside thefield of rational numbers, Q. The ring of integers is the model for all integraldomains. In this chapter we will examine integral domains in general, answeringquestions about the ideal structure of integral domains, polynomialrings over integral domains, and whether or not an integral domain can beembedded in a field.16.1 Fields of FractionsEvery field is also an integral domain; however, there are many integraldomains that are not fields. For example, the integers Z are an integraldomain but not a field. A question that naturally arises is how we mightassociate an integral domain with a field. There is a natural way to constructthe rationals Q from the integers: the rationals can be represented as formalquotients of two integers. The rational numbers are certainly a field. In fact,it can be shown that the rationals are the smallest field that contains theintegers. Given an integral domain D, our question now becomes how toconstruct a smallest field F containing D. We will do this in the same wayas we constructed the rationals from the integers.An element p/q ∈ Q is the quotient of two integers p and q; however,different pairs of integers can represent the same rational number. For instance,1/2 = 2/4 = 3/6. We know thatab = c dif and only if ad = bc. A more formal way of considering this problemis to examine fractions in terms of equivalence relations. We can think of277