Say I have two polynomials $f(x),g(x)\in\mathbb{Z}[x]$, both monic and irreducible. Denote by $K$ the splitting field of $f(x)$, and by $\alpha$ some root of $f(x)$. Is it true to say that if $g(x)$ is reducible over $K$ then it is reducible over $\mathbb{Q}[\alpha]$?
Thanks in advance!
No. Take $f=X^3-2$ so that $K=\mathbb{Q}(\sqrt[3]{2},j)$. Let $\alpha=\sqrt[3]{2}$, and let $g=X^2+X+1$. Then $g$ is reducible over $K$ since $g(j)=0$, but irreducible over $\mathbb{Q}(\sqrt[3]{2})$ since $g$ has no real roots.
Side remark. I don't see any connection between your question and your title.