Solving Complex Number Equation with Galois theory Suppose that the complex numbers $\alpha$, $\beta$ and $\gamma$ satisfy
\begin{align*}
  \alpha + \beta + \gamma &= 3, \\
  \alpha^2 + \beta^2 + \gamma^2 &= 5, \\
  \alpha^3 + \beta^3 + \gamma^3 &= 12.
\end{align*}
I want to show that $\alpha^n + \beta^n + \gamma^n \in \mathbb{Z}$ for all $n \in \mathbb{Z}^+$ if possible, using the concept of symmetric polynomials from Galois theory, but I am not sure how.

Use $(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)\iff ab+bc+ca=?p$(say)
$a^3+b^3+c^3-3abc=(a+b+c)\{(a+b+c)^2-3(ab+bc+ca)\}\iff abc=q$(say)
So, $a,b,c$ are the roots of $$t^3-3t^2+pt-q=0$$
$\implies t^{n+3}=3t^{n+2}-pt^{n+1}+qt^{n}$
If $S_m=a^m+b^m+c^m,$
$$S_{m+3}=3S_{m+2}-pS_{m+1}+qS_m$$
Now use Strong induction
