if the sum of two numbers $\alpha$ and $\beta$ is algebraic, and their product is transcendental, what do we know about these numbers?

Question

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!

Answer 1

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 $\beta = ((\alpha + \beta) - (\alpha - \beta))/2$.

Answer 2

Let $a=\frac{\alpha+\beta}2$ and $b=\frac{\alpha-\beta}2$. Then $a$ is algebraic, $\alpha=a+b$, and $\beta=a-b$. And $b$ cannot be algebraic because, otherwise, $a$ and $b$ would be both algebraic and therefore $ab$ would be algebraic too. So, $b$ is transcendental.