Find an example about splitting field

Since $E$ is a splitting field, it is a Galois extension of $\mathbf{Q}$. Thus $N=\mathrm{Gal}(K/E)$ is a normal subgroup of $G=\mathrm{Gal}(K/\mathbf{Q})$. Since $f$ is irreducible, the group $G$ acts transitively on its roots. Let $\alpha$ and $\beta$ be roots of $f$, and choose $g \in G$ with $g(\alpha)=\beta$. Using normality of $N$ in $G$, the stabilizers $N_\alpha$ and $N_\beta$ of $\alpha$ and $\beta$ in $N$ are related by $$g N_\alpha g^{-1}=N_\beta,$$ and in particular they have the same cardinality. Hence by the orbit stabilizer theorem the degrees over $E$ of $\alpha$ and $\beta$ are equal.