Exercise 15, § 3

Give an example of a field K containing two subfields K₁ and K₂ such that K is algebraic over K₁ and over K₂ but not over K₁ ∩ K₂. (Take K = Q(T), K₁ = Q(T²), and K₂ = Q(T² − T). Verify that [K : K₁] = 2, [K : K₂] = 2, and K₁ ∩ K₂ = Q. For this, observe that an element f(T) of K₁ ∩ K₂ satisfies the conditions f(−T) = f(T) and f(1 − T) = f(T).)