consider the field K = F(α+β,αβ) which is a finite extension of F since α+β and αβ are algebraic over F.

note that α,β are the roots of x^{2}- (α+β)x + αβ in K[x].

thus [F(α,β):F] = [F(α,β):K][K:F], since both factors are finite, so is their product.

thus the subfields F(α) and F(β) of F(α,β) must have finite degree so α,β are algebraic.