1.1 The Field Properties 9
(d) Exercise 12. •
Theorem 1.1. 8 (Properties of Division and Fractions) In any fi eld F ,
(a) Vx E F , if x -/= 0, then 0 ...;-x = O;
(b) Vx E F , x -;- 1 = x; if x-/= 0 , then 1-;-x = x-^1 ;
(c) Vx E F , if x-/= 0, then (-x)-^1 = - x-^1 [that is, - (x-^1 ) ] ;
x
(d) Ify-/=0,then-=0{.;:}x =O;
y
a ac
(e) Ifb,c-/=O, th en b =be;
a c ad+ bc
(f) If b, d-/= 0, then b + d = bd ;
a c ac
(g) If b, d -/= 0, then b · d = bd;
a - a a
(h) Ifb-/=O, then-b=b= -b;
(i) If a, b-/= 0, th en (~)-
1
= ~-
b
(j) If a-/= 0, then th e equation ax + b = 0 has th e unique solution x = --.
a
Proo f. (a ) Exercise 13.
(b) Exercise 14.
(c) Exercise 15.
(d) Let y -/= 0 in F. Then y-^1 -/= 0 by Theorem 1.1.4 (f). By definition,
~ = x · y-^1. Thus,
x
- = 0 {.;:} x. y -^1 = 0
y
{.;:} x = O or y -^1 = 0 by Theorem 1.1.4 (e)
{.;:} x = 0, since y -^1 -/= 0.
(e) Suppose b, c-/= 0. Then , since c · c-^1 = 1,
ac = (ac). (bc)-^1 by Definition 1.1.6
be
= (ac)(c-^1 b-^1 ) (give reasons)
a
= a(c · c^1 )b-^1 = a(l)b-^1 = -,;·
(f) Exercise 16.
(g) Exercise 17.