I'm having a hard time grasping the concept of a field of quotients. The book I'm currently reading gives the following definition:
Any integral domain $D$ can be enlarged to a field $F$ such that every element of $F$ can be expressed as a quotient of two elements of $D.$ Such a field $F$ is a field of quotients of $D.$
Can someone break this definition down into something much easier to understand and possibly provide me with an example?
The prototypical example is how the rational numbers are ratios between integers. The whole notion of fraction field is based off of abstracting this one example, so understand it.
If $D$ is a domain and $F$ a field containing it, then $F$ contains reciprocals of nonzero elements from the domain $D$, and so it contains all fractions $a/b$ with $a\in D,b\in D^\times$. As it turns out, these fractions suffice to form a field containing $D$, since fractions can be added, subtracted, multiplied and divided and we still get fractions. So the smallest field containing $D$ is the fraction field.