Let $K$ be the splitting field of a polynomial $f\in \mathbb Q[x]$, which is irreducible mod 3, with $G:=Gal(K|\mathbb Q)=S_n$ (symmetric group). Let $U$ be a subgroup of $G$ with fixed field $F:=fix(U)$ and write $F=\mathbb Q(\alpha)$.
Now let $x,y \in \mathbb N$ with $gcd(x,3)=1$ and $gcd(y,3)=1$.

Why is the polynomial $F_\alpha(X):=X^3-3X-6\alpha\frac{x}{y} \in F[X]$ irreducible over $K$?

It looks like an Eisenstein polynomial.

Is here ramification theory needed?