if the sum of two numbers $\alpha$ and $\beta$ is algebraic, and their product is transcendental, what do we know about these numbers? These are elements of a field. My intuition says that $\alpha=a+b$, $\beta=a-b$, where, $a$ is algebraic and $b$ is transcendental, but I can't prove it. I don't even know where to start.
Thanks in advance!

If $\alpha + \beta$ is algebraic and $\alpha \beta$ is transcendental, then
$(\alpha - \beta)^2 = (\alpha+\beta)^2 - 4 \alpha \beta$ is transcendental, so $\alpha - \beta$ is transcendental.
Thus $\alpha = ((\alpha + \beta) + (\alpha - \beta))/2$ is transcendental, and so is
$\beta = ((\alpha + \beta) - (\alpha - \beta))/2$.
