Let $F$ be a field and let $f(x)\in F[x]$. Denote the splitting field of $f(x)$ by $E$. Denote the roots of $f(x)$ by $\alpha_1,...,\alpha_n$.
A polynomial $g(x_1,...,x_n) ∈ F[x_1,...,x_n]$ gives an relation between the roots if $g(α_1,...,α_n) = 0.$
Now let $σ\in Aut_F(E)$, and let $τ$ denote its image in $S_n$.
Show that if $g(x_1,...,x_n) ∈ F[x_1,...,x_n]$ gives a relation between the roots, then so does the polynomial $g(x_{τ(1)},...,x_{τ(n)})$ obtained by permuting the variables by $τ$.
It seems to be obvious but I am now stuck on how to give a formal explaination. Could someone please help? Thanks in advance!
Since the coefficients of $g$ are in $F$ and $\sigma\in\mathrm{Aut}(E/F)$, it follows that $$ \sigma(g(\alpha_1,\dots,\alpha_n))=g(\sigma(\alpha_1),\dots,\sigma(\alpha_{n}))=g(\alpha_{\tau(1)},\dots,\alpha_{\tau(n)})$$
In particular, if $g(\alpha_1,\dots,\alpha_n)=0$ then $g(\alpha_{\tau(1)},\dots,\alpha_{\tau(n)})=\sigma(0)=0$ as well.