If $x$ and $y$ are complex numbers and $x+y$ , $xy$ are algebraic numbers then how to prove that $x$ and $y$ are also algebraic numbers?

Question

I tries basic operations like multiplication and addition in a hope that i will get $x$ and $y$ out of $x+y$ and $xy$ but that didn't worked for me.Also i tried assuming a polynomial with rational coefficients which has $x+y$ as solution and another polynomial with rational coefficients which has $xy$ as solution and tried to find a polynomial which has $x$ as solution .But i couldn't do that also .

Answer 1

Simce $(x-y)^2=(x+y)^2-4xy$, the numbers $x$ and $y$ can be written as $\dfrac{x+y\pm\sqrt{(x+y)^2-4xy}}2$. Since the square root of an algebraic number is again an algebraic number, since the sum (and the difference) of algebraic numbers is again an algebraic number and since half an algebraic number is again an algebraic number, this proves that $x$ and $y$ are algebraic.