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Lemma 5.4.1.
Let D be an integral domain, and let
W = { (a,b) | a,b belong to D and b 0 } .
The relation ~ defined on W by (a,b) ~ (c,d) if ad = bc is an equivalence relation.
Definition 5.4.2.
Let D be an integral domain.
The equivalence classes of the set
W = { (a,b) | a,b belong to D and b 0 }
under the equivalence relation defined by (a,b) ~ (c,d) if ad = bc will be denoted by [a,b].
Lemma 5.4.3.
For any integral domain D, the following operations are well-defined on Q(D).
For [a,b], [c,d] in Q(D),
[a,b] + [c,d] = [ad+bc,bd] and [a,b] · [c,d] = [ac,bd] .
Theorem 5.4.4.
Let D be an integral domain.
Then there exists a field F that contains a subring isomorphic to D.
Definition 5.4.5.
Let D be an integral domain.
The field Q(D) defined in Definitino 5.4.2 is called the
field of quotients or field of fractions
of D.
Theorem 5.4.6.
Let D be an integral domain, and let
: D -> F
be a one-to-one ring homomorphism from D into a field F.
Then there exists a unique extension
: Q(D) -> F
that is one-to-one and satisfies
(d) = (d) for all d in D.
Corollary 5.4.7.
Let D be an integral domain that is a subring of a field F.
If each element of F has the form ab^{-1} for some a, b in D,
the F is isomorphic to the field of quotients Q(D) of D.
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