Suppose $a$ and $b$ are complex numbers and both transcendental over $\mathbb Q$. I am wondering why $ab$ and $a+b$ can not both be algebraic.

Thanks for any help.


Hint: Suppose $s=a+b$ and $p=ab$ are both algebraic numbers. Then,


IOW, $a$ is the root of a second degree polynomial with algebraic coefficients.


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