Intermediate field extensions and Degree of field extension

Question

I was wondering whether there is a relation between $[L:K]$ and the number of intermediate fields $F$, where $K\subseteq F\subseteq L$. If there is then can someone please explain why. What if $L/K$ was Galois, would that make a difference to whether a relation exists or not.

Answer 1

In the special case of an finite extension $L/K$ which is Galois with Galois group $G$, there is a bijection between the set of the subgroups of $G$ and the set of the subfields of $L$ containing $K$. This makes it easy to see that the degree of the extension $L/K$ and the number of subfields of $L$ containing $K$ do not determine each other.

For example, if $p$ is a prime and $n\geq1$, there exists Galois extensions $L/K$ of group $\mathbb Z/p^n\mathbb Z$, which have degree $p^n$ and have $\phi(p^n)=p^{n-1}(p-1)$ subextensions, and there exist Galois extensions $L/K$ of group $(\mathbb Z/p\mathbb Z)^n$, which have degree $p^n$ and $$\dfrac{(p^n-1)(p^n-p)\cdots(p^n-p^{k-1})}{(p^k-1)(p^k-p)\cdots(p^k-p^{k-1})}$$ subextensions. In fact, there are many, many non-isomorphic groups of order $p^n$, and one probably gets a lot of different numbers of subgroups.