Bridge to Abstract Mathematics: Mathematical Proof and Structures

9.1 FIELDS 297

EXAMPLE 4 For those familiar with a little matrix theory, the set of all

2 x 2 matrices, with operations of ordinary matrix addition and multi-

plication, provides us with an algebraic structure satisfying a number

of the field axioms. The additive part of this structure, in fact, satisfies

field axioms 1 through 5, with ( i) being the additive identity and

( 1:) the additive inverse of the matrix ( :). Furthermore,

matrix m~iti~lication distributes over addition, is associative, and has

an identity, namely, ( ) The structure fails to be a field on two

counts: Matrix multiplication is noncommutative and not every 2 x 2

matrix has a multiplicative inverse. In fact, the matrices having inverses

are precisely those with nonzero determinant ad - bc.

EXAMPLE 5 For any positive integer m, denote by Z, the set of symbols
(0, 1,2,... , m - 1). We define operations "plus" and "times" on such
sets as follows: if a, b E Z,, we calculate the "sum" a + b, in Z,, by com-
puting the ordinary sum of a and b; that is, their sum in Z, and writing
down the remainder (necessarily an integer between 0 and m - 1, in-
clusive. Recall the "division algorithm for Z," stated in the solution to
Example 10, Article 6.3) upon division of that ordinary sum by m. Thus
in Z,, 7 + 7 = 6, 2 + 7 = 1, and 4 + 4 = 0. The calculation of the
"product" in Z, is totally analogous, replacing "ordinary sum" by
"ordinary product." Thus, in Z,, we have 7 - 7 = 1,2 7 = 6, and 4 4 =

0. You should construct the complete "addition" and "multiplication"
   tables for Z, and use them to check the various field axioms. We recom-
   mend special attention to the associative, identity, and inverse axioms,
   as well as distributivity of multiplication over addition (the fact that
   the latter works in all cases is rather remarkable). In the course of doing
   this you should discover that (Z,, +, -) is not a field.
   We list in Figure 9.2 complete addition and multiplication tables for
   (Z,, +, 0) and (Z,, +, 9). Both structures satisfy all field axioms except
   possibly Axiom 10, the multiplicative inverse axiom. Now (Z,, +, a)
   satisfies Axiom 10 also, and so is a field. The criterion determining
   whether a structure (Z,, +, .) is a field is suggested by Exercise 2, Article
   6.2. Such a structure satisfies Axiom 10, and so is a field, if and only if
   m is prime. Structures (Z,, +, -) are called the integers modulo m.

EXAMPLE 6 Consider the set of those real numbers having the form

a + b&, where a and b are rational. Let us denote this subset of R

(and superset of Q (Why?)) by Q(&). Exercise 11 calls for you to pro-

vide portions of the proof that Q(&) is a field, under the ordinary

addition and multiplication it inherits from R.