Solving Complex Number Equation with Galois theory

Question

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.

Answer 1

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

Answer 2

Use Newton-Girard's relations for symmetric polynomials: if

$$\sigma_1=x_1+\dots+x_n,\enspace\sigma_2=\mkern{-12mu}\sum_{1\le i<j\le n}\mkern{-12mu}x_i x_j,\enspace\sigma_3=\mkern{-18mu}\sum_{1\le i<j<k\le n}\mkern{-18mu}x_i x_j x_k, \;\dots\dots,\enspace\sigma_n=x_1\dotsm x_n,\enspace\sigma_k=0\enspace\text{if }k>n, $$ are the elementary symmetric polynomials, and $$s_k=x_1^k+\dots+x_n^k,$$ are the Newton sums, one has the relations: $$k\sigma_k=\sum_{i=1}^n(-1)^{i-1}\sigma_{k-i}\,s_i .$$ These relations can be used for an inductive proof.