Bridge to Abstract Mathematics: Mathematical Proof and Structures

298 PROPERTIES OF NUMBER SYSTEMS Chapter 9

10.

I

II.

the field, satisfying 1. a = a for

all a E F

To each a E F, there corresponds an

an element b E F having the property

that ab = ba = 1. The element b,

which can be shown to be uniquely

determined by a (see Theorem I), is

denoted a-' and called the reciprocal

of a

If a, b, c E F, then a(b + c) = ab + ac

(multiplicative identity axiom)

(multiplicative inverse axiom)

(multiplication distributes over

addition)

Since 1 # 0 in a field, any field must contain at least two elements. Also,

Axioms 1 and 6 are, strictly speaking, not necessary, due to the definition

of binary operation (Definition I), which already forces closure. We include

them, nonetheless, for the sake of emphasis.

The set of real numbers, with the usual operations of addition and multi-

plication, constitutes a field, by its definition as the (unique) complete

ordered field and our assumption in this chapter that such an object exists.

Therefore the fact that addition and multiplication of real numbers satisfy

the field axioms, a fact that is familiar to any high school algebra student,

is dictated by the definition of R; in particular, it is not something that has

to (or can) be proved. All other familiar properties of R are either additional

axioms involved in the definition of "complete ordered field (to be studied

later in the chapter) or are theorems that can be proved from the axioms

for a complete ordered field. R is often thought of as the prototypical field,

and the borrowing of real number notation, for example, +, ., 0, 1, -a, and

so on, to denote aspects of general fields, bears this out. But the concept

of abstract, or generic, field is much more general than just the real numbers,

as the following examples show.

EXAMPLE 2 (Q, +, .) and (C, +, .) are both fields. Looking at some key

axioms, we note that the rational numbers 011 and 111 are the additive

and multiplicative identities in Q, whereas 0 + Oi and 1 + Oi play those

roles, respectively, in-C. If a/b is a nonzero rational, then b/a is its

multiplicative inverse; if a + bi is a nonzero complex number, then

(a - bi)/ (a2 + b2) is its multiplicative inverse. Since Q is a subset of R

and is itself a field under the two operations "inherited" from R, Q is

said to be a subfield of R. Properties of Q and C will be discussed in

some detail later, C in Article 9.4 and Q in Chapter 10. Cl

EXAMPLE 3 The substructures (N, +, a) and (Z, +, .) of the real number

field fail to be fields. For example, N fails to satisfy Axiom 4, among

others, whereas Z violates Axiom 10. You should determine which of

the field axioms are satisfied by these two algebraic structures. Cl