I need to show that if the sum and the product of two complex numbers is algebraic, then each of then is algebraic.
We have that the extensions $Q(a+b)/Q$ and $Q(ab)/Q$ are finite, so the extension $Q(a+b,ab)/Q$ is finite.
Since $$[Q(a,b):Q]=[Q(a,b):Q(a+b,ab)] [Q(a+b,ab):Q],$$ it suffices to show that the extension $Q(a,b)/Q(a+b,ab)$ is finite, but I can't come up with an appropriate basis. Any ideas?