Thread: Apostol's Field Axioms View Single Post
 Sci Advisor P: 906 Apostol's Field Axioms i'm not sure which "axiom" you're talking about, is it: 1/(xy) = (1/x)(1/y)? one doesn't need "uniqueness" of an inverse to show that, just "existence": (xy)[(1/x)(1/y)] = x(y[(1/x)(1/y)] (associativity) = x(y[(1/y)(1/x)] (commutativity) = x([y(1/y)](1/x)] (associativity) = x((1)(1/x)) (using y(1/y) = 1, since 1/y is "a" inverse for y) = x(1/x) (multiplicative identity property of 1) = 1 (since 1/x is "a" inverse for x). thus 1/(xy) is "one" inverse for xy (it might be, that there are others, but at the very least we know for sure that (xy)(1/(xy)) = 1 if x(1/x) = 1, and y(1/y) = 1). what is going on is this: the actual process of computing a/b + c/d goes like this: 1. a/b + c/d = (a/b)(1) + (c/d)(1) = 2. (a/b)(d/d) + (c/d)(b/b) = 3. (ad/(bd)) + (bc/(bd)) = 4. (ad + bc)/(bd) 1-->2 is obvious 2-->3 uses commutativity and associativity of multiplication (which is why writing 1/x = x-1 makes it clearer) 3-->4 uses the distributive law for the above to work, we just need the existence of a-1 and b-1, we don't need uniqueness (the possibility of division). of course, uniqueness is easy to prove given existence: suppose ab = ba = 1, and ca = ac = 1: then c = c(1) = c(ab) = (ca)b = (1)b = b.