Recently I learned about Pisot-Vijayaraghavan numbers, and proofs around them led me to the fact that the sums of $n$th powers of roots of a monic irreducible integer polynomial are integers. I saw a proof, but it used power series, and I thought it must be provable with Galois Theory, so I came up with this shorter proof:
Let $p=\sum_{i=0}^np_ix^i$, $p_n\neq0$ be a monic irreducible integer polynomial with roots $\alpha_1$ to $\alpha_n$, and let $f\in\mathbb{Z}[\alpha_1,...\alpha_n]$ be a symmetric expression in the roots. Then $f$ evaluates to an integers.
By Guass's Lemma we know that $p$ is irreducible in $\mathbb{Q}$ and thus separable. So its splitting field $\mathbb{Q}_f$ is Galois. $f$ is fixed by all automorphisms of $\mathbb{Q}_f$ since they all must fix the set of roots of $p$. Hence, $f\in\mathbb{Q}$. But since algebraic integers form a ring $f$ is an algebraic integer, say a root of integer polynomial $q$. Now $q$ factors in $\mathbb{Q}$ and by Gauss's Lemma, in $\mathbb{Z}$. One of these factors must have root $f$, so we apply again, until we obtain that $x-f$ is an integer polynomial, so $f\in\mathbb{Z}$.
I was wondering how far this could be generalised. I know it works if we replace $\mathbb{Z}$ with an integral domain of characteristic $0$, and $\mathbb{Q}$ with its field of fractions. However, I was wondering if we could drop the characteristic $0$ condition. My proof does not generalise since it relies on irreducible polynomials being separable.