298 PROPERTIES OF NUMBER SYSTEMS Chapter 9
Figure 9.2 (a) Tables defining the field of integers modulo 3; (b) tables
defining the algebraic structure, the integers modulo 4.
EXAMPLE 7 Denote by R[x] the set of all polynomials in a single vari-
able x having real coefficients. Endow R[x] with the operations of or-
dinary polynomial addition and multiplication. It can be shown that
(R[x], + ,. ) fails to be a field only in that it violates Axiom 10. Fur-
thermore, there is a field associated with R[x] in a canonical way (ac-
tually a field that can be constructed from R[x], by means of standard
type of construction that we will study in Chapter 10). This associated
field is denoted R(x) and may be thought of as consisting of the rational
functions, that is, quotients of polynomials with real coefficients.
Here is an important distinction between the role of R in this chapter
and that of the structures presented in Examples 2 through 7. Under our
approach here, where we assume a complete ordered field exists, R is defined
- to be a field. Hence the 11 field properties are axioms for R, in particular,
there is no question in this chapter of proving or verifying any of these
properties for R. On the other hand, each of these structures is defined
independently of the field concept. The fact that each is or is not a field is
a theorem, that is, each of the field axioms must be verified or demonstrated
to be false for each of those examples. In Chapter 10, where we drop the
aforementioned existence assumption and outline a proof of the existence
of a complete ordered field, the approach is to construct a mathematical
object and prove it is a complete ordered field. In that approach the role
