Let $L/K$ be a field extension and $a,b\in L $. It can easily be shown that if $a,b$ are algebraic over $K$ then sum and product are, too.
But I read that also the converse is true, say if $a+b$ and $ab$ are algebraic over $K$ then so is $a$ and $b$.
My strategy is to use that, since product and sum are algebraic we have that
$\displaystyle [\mathbb{Q}(ab,a+b):\mathbb{Q}]$ is finite, thus algebraic.
Now by using field operations (denoted $f$, e.g. multiplication, inverse ...) I somehow have to obtain an expression $a=f(a+b,a*b)$ and the same for $b$. Then the conclusion would follow. However, I could not achieve this.
Hint : $(X-a)(X-b) =X^2 - (a+b)X + ab$