Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

§ 5.4 Quotient Fields

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].
The set of all such equivalence classes will be denoted by Q(D).

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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