Chapter 5.3, Problem 10E

Since this section presents a method for constructing a field of quotients for an arbitrary integral domain $D$, we might ask what happens if $D$ is already a field. As an example, consider the situation when $D={\mathbb{Z}}_5$.

a. With $D={\mathbb{Z}}_5$, write out all the elements of $S$, sort these elements according to the relation $\sim$, and then list all the distinct elements of $Q$.

b. Exhibit an isomorphism from $D$ to $Q$.