Sum and product of two transcendental numbers can't be both algebraic

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.

• $x^2-(a+b)x+ab=(x-a)(x-b)$. (and the set of algebraic numbers is an algebraically closed field.) – symplectomorphic May 3 '14 at 3:44
• Do you mean real ? – Claude Leibovici May 3 '14 at 4:14
• – Martin Sleziak Aug 16 '14 at 13:22

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.