Let a,b complex numbers. Prove if a+b ,ab are algebraic then a,b are also algebraic?

Answer 1

if S = a+b and P= ab
then a and b are the roots of the equation:
f(x) = x^2 - Sx + P= 0
I assume you say algebraic over Q.
Now S,P are algebraic over Q implies that the roots of f are algebraic over Q, i.e a and b.
why? because a and b are algebraic over the field of algebraic numbers so they become algebraic too.

Answer 2

let
a=x+iy
b=x-iy
a+b=2x
a-b=2iy
ab=(x+iy)(x-iy)=x^2 -y^2