property of roots I need to prove $\gamma_n$ is an integer for any value of $n$. $\gamma_n$ is defined as $= \alpha^n + \beta^n$. $\alpha$ and $\beta$ are roots of equation $x^2 + mx - 1 = 0$, $m$ is an odd integer.
I tried by induction, but do not know how to prove $\gamma_n$ is an integer for $n + 1$. Can some body help

$\alpha$ and $\beta$ are algebraic integers, hence so is $\alpha^n+\beta^n$ for each $n$. Moreover, $\alpha^n+\beta^n$ is fixed by $\mathrm{Gal}(\mathbb{Q}(\alpha)/\mathbb{Q})$, so is a rational number. And if a rational number is integral over $\mathbb{Z}$, then it's an integer.
