Show that if F, E, and K are fields with F ≤ E ≤ K, then K is algebraic over F if and only if E is algebraic over F, and K is algebraic over E. (You must not assume the extensions are finite.)

Let i, r, and k be positive integers, and let s gcd(i, r). Assume that r | ik. Prove that | k.

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].

Let a, b ∈ Z, not both zero. Prove gcd(−a, b) = gcd(a,−b).