Is the degree of splitting field remains same when we shift to an algebraic extension?

Question

Question from my own curiosity:

Let $F$ be any field and $f(x)\in F[x]$ be an irreducible polynomial with a splitting field $L$ over $F$. Now suppose $K/F$ be any algebraic extension and $f$ is irreducible over $K$ also. My question is whether the degree of extension of any splitting field of $f$ over $K=|L:F|$?

Answer 1

No. Let $\zeta=e^{\frac{2\pi i}{3}}$. The polynomial $x^3-2$ is irreducible over both $\mathbb{Q}$ and $\mathbb{Q}(\zeta)$. But the extension degree of the splitting field (which is $\mathbb{Q}(\sqrt[3]{2},\zeta)$ over both fields) in not the same.