Bridge to Abstract Mathematics: Mathematical Proof and Structures

9.1 FIELDS 301

Analogs to the familiar subtraction and division operations from R
exist in an arbitrary field.

DEFINITION 3

Let (F, +, .) be a field with x, y, E F. We define:

(a) The difference x- y (x minus y) by the rule x- y= x + (-y)

(b) The quotient x/y (x divided by y) by the rule x/y = xy- ', where y # 0

The restriction "division by nonzero divisors only" results from the fact
that 0-' does not exist in any field (a consequence of Theorem 2, see
Exercise 3). Subtraction and division in a field have properties that are
familiar from experience with real numbers. Several such properties are
grouped together in the following theorem.

THEOREM 5

Let (F, +, .) be a field with w, x, y, z E F. Then:

If w # 0, then w-I = l/w

If x # 0, then (w/x)-' = x/w

x = y if and only if x - y = 0

x/y = 1 if and only if x = y and y # 0

w-(X-y)=(w-x)+y

-w-x= -(w+x)

If x # 0 and y # 0, then xwlxy = w/y

If x # 0 and z # 0, then (wlx) + (y/z) = (wz + xy)/(xz)

If x # 0 and z # 0, then (wlx). (y/z) = wylxz

Partial proof (g) By the definition of a quotient, xwlxy = (xw)(xy)-' =

(xw)(x- l y - I) = (xx- l)(wy - l) = wy - ' = wly. The step (xw)(xy)- ' =

(xw)(x- 'y - ') follows from Exercise 6(a).

(h) By (g), (wlx) + (yl4 = (wzlxz) + (xylxz) = (wz + xy)l(xz). The

final equality follows from Exercise 8(a). The remaining parts of the

proof are left as exercises.

Part (c) of Theorem 5 can be combined with Theorem 4 to provide a

proof of the multiplicative cancellation property of a field. This proof is left

as an exercise [Exercise 8(d)].

THEOREM 6

If (F, +, .) is a field, if x, y, z E F with xy = xz and x # 0, then y = z.

Let us test parts of Theorem 5, using the field Z,. In (b), let w = 3 and

x = 4. Then x- ' = 2, whereas w -I = 5. Hence w/x = wx- ' = (3)(2) = 6,

so that (w/x)- l = (6)- ' = 6. On the other hand, x/w = xw - ' = (4)(5) = 6.

Hence (w/x)- ' = 6 = xlw, as predicted by (b) of Theorem 5. In (e), let w = 1,

x=5, and y=6. Then x-y=x+(-y)=5+(-6)=5+1=6, so