Math 554- 703 I - Analysis I
Solutions for Problems 1-2 of Homework Assignment # 1
1. Using the field axioms, prove that for each aโF, a 6= 0, the multiplicative
inverse of ais unique.
Solution: Let abe an arbitrary member of Fand suppose that ahas multiplicative inverses band c, that is
a b = 1 (1)
a c = 1 (2)
In this case, we can multiply both sides of equation (1) by cto obtain
(a b)c= 1 c
Using the field axioms, this reduces to
b(a c) = c
and so b=cusing equation (2) and the identity axiom that b1 = b.2
2. Using any of the results we proved in class (before the statement of this result)
carefully prove that (-1)(-1)=1.
Solution: We know from our previous results which were established in class that the additive inverse of the additive
inverse of an element is the element itself
โ(โa) = a. (3)
We also know for each aโFthat
(โ1) b= (โb)
the additive inverse of b. Therefore applying this with b=โ1 and using equation (3) we obtain
(โ1)(โ1) = โ(โ1) = 1 2