If $\gamma$ and $\overline\gamma$ solve the equation $z^2+az+b=0$ and $\gamma$ is not a real number (complex number or just an imaginary number), does it mean that necessarily $a$ and $b$ are real numbers?
same with $z^3+az^2+bz+c=0$, can $a$, $b$ and $c$ necessarily be real numbers?
Thank you!
Regarding the quadratic case:
Say $f(x) = x^2 + ax + b$ factors as $f(x) = (x-\alpha)(x- \overline{\alpha})$. Re-expanding, we get $f(x) = x^2 - (\alpha + \overline{\alpha})x + \alpha \overline{\alpha}$. Because $\alpha$ and $\overline{\alpha}$ are complex conjugates, their sum has an imaginary part of $0$. Likewise, you can check that $\alpha \overline{\alpha} = |\alpha|^2$, which is real.
You could use a similar approach to investigate the cubic. Note also that if a cubic has complex roots but real coefficients, then exactly one of its roots must be real because complex roots of polynomials in $\mathbb{R}[x]$ come in conjugate pairs.
In generality, the coefficients of a monic polynomial are elementary symmetric polynomials evaluated at its roots; this fact is of great importance in the theory of equations and Galois thoery.