Field of fractions
Given an integral domain, , we may informally define the field of fractions of (also called the fraction field or the quotient field), denoted by , as the set . This is analogous to the construction of the rational numbers from the integers, , and can be viewed as a way turning into a field.
While the above definition makes sense intuitively, it is not entirely satisfactory. In general, division in is undefined, so an expression like is meaningless.
To get around this, we consider the set of ordered pairs and define an equivalence relation, , on by if . Then we can define as the set of equivalence classes of under .
Intuitively we can think of each ordered pair as representing the fraction , and our definition of is equivalent to the statement
We can now define addition and multiplication on in the 'obvious way':
Notice that we have actually defined these operations on , not on . However it is now easy to verify that if and for then and , so we can view these operations as operations on . It is now a simple matter to verify that is indeed a field under these operations.
We can view as a subring of via the embedding . We can now think of as the 'smallest' field which contains .
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