Suppose I'm given a field extension $K\subset L\subset K(\alpha)$ with $[K(\alpha):K]=d\in \Bbb{N}$. Then find all possible values of the degree $[L(\alpha):L]$. It's easy to observe that the minimal polynomial of $\alpha$ over $L$ divides the minimal polynomial of $\alpha$ over $K$ and hence $[L(\alpha):L]\le d$. Can someone give an example such that there are intermediate fields for which all possible values are achieved?
Edit:
Can this example works? Let $[K(\alpha):K]=d$ where $K=\Bbb{Q}$ and $\alpha$ is a root of $X^d-2$. By Eisenstein $X^d-2$ is irreducible over $\Bbb{Q}$. Then for any divisor $d'$ of $d$, we consider $L=\Bbb{Q}(\beta)$ where $\beta$ is a root of $X^{d'}-2$. Since $\beta=\alpha^{d/d'}$, $L\subset \Bbb{Q}(\alpha)$ and $[L(\alpha):L]=\frac{[K(\alpha):K]}{[L:K]}=\frac{d}{d'}$. Thus we can achieve all divisors of $d$ as degrees of intermediate fields. Is this description correct?