Suppose $M : L$ and $L : K$ are extensions, and that $\alpha \in M$ is algebraic over $K$. Does $[L(\alpha):L]$ always divide $[K(\alpha):K]$?

Question

Supposing that $\alpha \in M$ is algebraic over $K$, we then know that $[M(\alpha):L]<\infty$ and $[L(\alpha):K]<\infty$ using the tower law.

How can I find out about $[K(\alpha):K]$ or $[L(\alpha):L]$ from this information?

Answer 1

Consider $K = \mathbb{Q}$, $L = \mathbb{Q}(\sqrt[3]2)$, and $\alpha = \zeta\sqrt[3]2$ where $\zeta$ is a primitive cube root of unity.

Then $[L(\alpha):L] = 2$ and $[\mathbb{Q}(\alpha):\mathbb{Q}] = 3$.