3. True or False: If F is a field, the units of Fx] are precisely the nonzero elements of F. (a) True b) False

identify the statement that does not hold in a complete ordered field K and provide a counterexample, or an argument as to why it does not hold. it's Z ⊂ K in this case. (i) ∀b ∈ K, ∃a ∈ K, 4ab = b. (ii) ∀a ∈ K, ∃b ∈ K, [(6ab = b) ∧ (b ≠ 0)]. (iii) ∀a∈ K, [0 < a < 1 ⇒  a2 < a].

Show that the set Q + Q √ 3 is a field. Q denotes set of all rational numbers

Theorem 1.2.17 (Intervals) In an ordered field F, the following sets are intervals: (a) [a, b] = {x E F:a ≤x≤b}; (This could be {a} or Ø.) (b) (a, b) = {x E F:a < x