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?
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$\begingroup$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.
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