Math 3210-3
HW 8
Due Tuesday, September 18, 2007
Ordered Fields
1. ♣ Let x, y ∈ R. Prove xy = 0 iff x = 0 or y = 0.
2. ♣ Let x, y, z ∈ R. Prove that if x < y and z < 0, then xz > yz.
3. Let x, y ∈ R. Prove |x · y| = |x| · |y|.
4. (a) Prove that |a + b + c| ≤ |a| + |b| + |c| for all a, b, c ∈ R.
(b) Use induction to prove |a1 + a2 + a3 + · · ·+ an | ≤ |a1 |+ |a2 |+ · · ·+ |an | for n numbers a1 , a2 , . . . , an .
5. ♣ Prove that in any ordered field F , a2 + 1 > 0 for all a ∈ F . Conclude from this that if the equation
x2 + 1 = 0 has a solution in a field, then that field cannot be ordered. (Thus, it is not possible to define
an order relation on the set of all complex numbers that will make it an ordered field.)