《Discrete Mathematics II教学-华南理工》Section 18 The Field of Quotients of an Integral Domain.pdfVIP

IV.21 Field of Quotients 1 Section IV.21. The Field of Quotients of an Integral Domain Note. This section is a homage to the rational numbers! Just as we can start with the integers Z and then “build” the rationals by taking all quotients of integers (while avoiding division by 0), we start with an integral domain and build a field which contains all “quotients” of elements of the integral domain. This is our first encounter with the idea of starting with an algebraic structure and then extending it to a larger, more complete structure. In this case we are extending an integral domain to a field that contains all inverses of elements of the integral domain (and possibly [probably] more). Note. We start with integral domain D and extend it to a field of quotients F following the text’s steps: Step 1. Define the elements of F . Step 2. Define + and · on F . Step 3. Verify the field axioms for + and · on F . Step 4. Show that F can be viewed as containing D as an integral subdomain. IV.21 Field of Quotients 2 Note. For part of Step 1, we define the set S = {(a,b) | a,b ∈ D,b = 0}. The analogy with Q is that we think of p/q ∈ Q as (p,q ) ∈ Z × Z. Notice that for p /q ,p /q ∈ Q if we have p /q = p /q then p q = p q . This is the motivation 1 1 2 2 1 1 2 2 1 2 2 1 for the next definition (and notice that equality of the “quotients” is dealt with in terms of multiplication). Definition 21.1. Two elements (a,b), (c,d) ∈ S are equivalent, denoted (a,b) ∼ (c,d), if and only if ad = bc. Lemma 21.2. The relation ∼ between elements of S is an equivalence relation. Note. To complete Step 1, we define F as