Let . both and are fields, , . If and are algebraic over , show that and are algebraic
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 x2 - (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.