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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.

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

$$p=ab=a(s-a)=sa-a^2,$$

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

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