1. Prove that in any ordered field F, a 2 + 1 > 0 for all a ∈ F. Conclude from this that if the equation x 2 + 1 = 0 has a solution in a field, then that field cannot be ordered. (Thus it is not...

1. 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.

2. Let S = {a, b} and define two operations ⊕ and ⊗on S by the following charts:

(a) Verify that S together with ⊕ and ⊗satisfies the axioms of a field.

(b) Identify the elements of S that are “0,” “1,” and “–1.”