is the sum of roots of a quadratic with rational coefficients always rational

Question

quadratic is $ax^2 + bx +c = 0$

let the roots be $f$ and $g$

as $f + g = -\frac{b}{a}\ $ and $\ f \cdot g = \frac{c}{a}$

does this imply if a quadratic has rational coefficients the sum of the roots and the product of the roots are also rational?

Answer 1

The same holds for higher-degree polynomials.

Consider the following: http://en.wikipedia.org/wiki/Vieta%27s_formulas. There are many relations between the roots of a polynomial and its coefficients, and as you might imply that's quite "magical": you can have a rational/integral quadratic with complex/real solutions, but sum and product of those are both rational/integral again.