
algebraic numbers
Let $\displaystyle F\subset{E}$. both $\displaystyle E$ and $\displaystyle F$ are fields, $\displaystyle \alpha$,$\displaystyle \beta\in{E}$. If $\displaystyle \alpha+\beta$ and$\displaystyle \alpha\beta$ are algebraic over $\displaystyle F$, show that $\displaystyle \alpha$ and $\displaystyle \beta$ are algebraic

Re: algebraic numbers
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.

Re: algebraic numbers