Answer Happy

1. Consider a general field an element x and the element -x which satisfies A5. Prove that the element -x is unique using the statements found in definition 1.1.5. (hint: show that if x +y = 0, then y must equal – x.)

1.1. BASIC PROPERTIES 23 Definition 1.1.5. A set F is called a field if it has two operations defined on it, addition x+y and multiplication xy, and if it satisfies the following axioms: (A1) If x E F and y EF, then x+yEF. (A2) (commutativity of addition) x+y=y+x for all x,y E F. (A3) (associativity of addition) (x+y)+z=x+(y+z) for all x, y, z E F. (A4) There exists an element 0 EF such that 0+x=x for all x EF. (A5) For every element x E F, there exists an element -x E F such that x+(-x) = 0. (M1) If x E F and y EF, then xy E F. (M2) (commutativity of multiplication) xy = yx for all x,y E F. (M3) (associativity of multiplication) (xy)z = x(yz) for all x,y,z E F. (M4) There exists an element 1 E F (and 1 + 0) such that 1x = x for all x E F. (M5) For every x E F such that x #0 there exists an element 1/x E F such that x(1/x) = 1. (D) (distributive law) x(y+z) = xy + xz for all x,y,z EF.