Sum and product of algebraic numbers

Question

I have encountered the following problem:

Let $L/K$ be a field extension and let $\alpha, \beta \in L$. Show that $\alpha$ and $\beta$ are algebraic over $K$ if and only if $\alpha + \beta$ and $\alpha\beta$ are algebraic over $K$.

For the forward implication I have already seen several ways to show it, but I cannot think of a way for the other direction.

Answer 1

Since $a$ and $b$ are the roots of the polynomial $$ f(x)=x^2-(a+b)x+ab, $$ $[K(a,b):K(a+b,ab)]$ is finite. Hence we have $$ [K(a,b):K]=[K(a,b):K(a+b,ab)] \cdot [K(a+b,ab):K]. $$ Assuming that $ab$ and $a+b$ are algebraic, the right hand side is finite, and hence also the left hand side, so that $a,b$ are algebraic.

Answer 2

$\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$

$(\alpha-\beta)^2=\alpha^2+\beta^2-2\alpha\beta$

$\alpha=\dfrac{\alpha+\beta+\alpha-\beta}{2}$

$\beta=\dfrac{\alpha+\beta-(\alpha-\beta)}{2}$