I recently came across the following claim.
Let $F$ be a field of characteristic $0$. Let $f\in F[X]$ have roots $y_1, \ldots , y_d$ in the algebraic closure of $F$. Define
$$ g_h = \prod_{1\le \lambda < \mu \le d} (X - y_\lambda - y_\mu -h y_\lambda y_\mu)$$
where $h \in \mathbb{Z}$.
The claim is that since $g_h$ is symmetric in $y_1, \ldots , y_d$, $g_h \in F[X]$. Why is this?
I think this question is straightforward but it has been awhile since I studied symmetric polynomials, Galois theory, etc. Any help is appreciated, as well as references. Thank you!
Let $K$ be the field $F(y_1, \ldots , y_d)$. Since $K$ is the adjunction to $F$ of all the roots of a polynomial in $F[X]$, $K$ is a splitting field over $F$. Since we are in characteristic $0$, this means that $K$ is Galois over $F$. In other words, if $x \in K$, and $\sigma x = x$ for every $F$-automorphism $\sigma$ of $K$, then in fact $x \in F$.
If $\sigma$ is any $F$-automorphism of $K$, then $\sigma$ permutes the elements $y_1, \ldots , y_d$ among themselves. The coefficients of $g_h$ are symmetric functions of the elements $y_\lambda + y_\mu + h y_\lambda y_\mu$, so the coefficients of $g_h$ are fixed by $\sigma$. So these coefficients must be in $F$, i.e. $g_h \in F[X]$. For example, the constant term of $g_h$ is $$c_0 = \pm \prod\limits_{\lambda, \mu} (y_{\lambda} + y_\mu + h y_\lambda y_{\mu})$$ and you can see that $\sigma c_0 = c_0$ for every $\sigma \in \textrm{Gal}(K/F)$.