Consider elements .
By definition, if for some .
Thus, since , and is an integral domain then either or is .
Hence, is an integral domain.
Now we can show that every non-zero element has an inverse.
Consider such that for .
Then,
so and we know that since .
Therefore, is a field so the nomenclature is reasonable.
Examples
General field
For let be a field, we take .
Now for any element , we can note that since
we can take .
Therefore, we can identify , which makes sense, since is already a field, so it already has all the units it needs.
Integers
For , note that are elements (equivalence classes) of the form for for .
This is the same as the definition for the rationals, .