Bridge to Abstract Mathematics: Mathematical Proof and Structures

9.1 FIELDS 295

multiplication of n x n matrices, and composition of mappings from a

given set X into itself. An operation may be defined on a finite set; this

is customarily done by means of a finite "multiplication table" such as

the one in Figure 9.1.

A set, having one or more (usually one or two) operations, constitutes
what is known as an algebraic structure. A Jield is an algebraic structure
with two operations.

Figure 9.1 Typical table used to specify a

particular jnite algebraic structure. What is the

relationship between this table and a table

in Figure 9.2a?

*

a

c

DEFINITION 2

Afield (F, +, .) consists of a nonempty set F, together with two binary operations

on F, denoted by the symbols " +" (plus) and ". " (times), satisfying the following

11 axioms:

I. If a, b E F, then a + b E F (additive closure)

2. If a, 6, c E F, then
   (a+b)+c=a+(b+c) (addition is associative)


3. If a, b, E F, then a + b = b + a (addition is commutative)


4. There exists an element in F,
   denoted "0" and called
   the zero, or zero element,
   of the field, satisfying
   a+O=O+a=a,
   for all a E F (additive identity axiom)


5. To each a E F, there corresponds an
   element b E F having the property that
   a + b = b + a = 0. The element 6,
   which can be shown to be uniquely
   determined by a (see Theorem I), is
   denoted -a and called minus a


6. Ifa,b~F,thenab~ F

a

a

bbc

c

7. If a, 6, c, E F, then (ab)c = a(bc)


8. If a, b E F, then ab = ba


9. There exists a nonzero element in F
   denoted "1" and called the unity of

(additive inverse axiom)

(multiplicative closure)

(Note the convention of

writing ab for a. 6.)

(multiplication is associative)

(multiplication is commutative)

b

a

C

a

b