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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6$x^2-(a+b)x+ab=(x-a)(x-b)$. (and the set of algebraic numbers is an algebraically closed field.) – symplectomorphic May 03 '14 at 03:44
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Do you mean real ? – Claude Leibovici May 03 '14 at 04:14
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See also http://math.stackexchange.com/questions/899097/sum-and-product-of-two-transcendental-numbers-cannot-be-simultaneously-algebraic – Martin Sleziak Aug 16 '14 at 13:22
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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.
David H
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