Let $K$ be the splitting field of a polynomial $f\in \mathbb Q[x]$, which is irreducible $\mod 3$, with $G:=\text{Gal}(K|\mathbb Q)=S_n$ (symmetric group). Let $U$ be a subgroup of $G$ with fixed field $F:=\operatorname{fix}(U)$ 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 $F_\alpha(X):=X^3-3X-\dfrac{6\alpha x}{y} \in F[X]$ irreducible over $K$?

It looks like an Eisenstein polynomial.

Is here ramification theory needed?