# Intermediate Field Extension Tower Property

my professor said the following was straightforward, and I can't figure it out(in fact, I'm not even sure why it should be true).

Suppose that $L/F$ is a field extension that can be decomposed into a finite tower of Galois extensions of degree $p$, where $p$ is prime. That is, $$F=L_0\subset L_1\subset ...\subset L_n=L$$ where $L_i/L_{i-1}$ is Galois with degree $p$ for $i>0$.

The question is this. If $F\subset K\subset L$ is an intermediate field, then does $K/F$ also have this tower property? It's not too hard to show that this is true if I assume that $K/F$ is Galois(using the fundamental theorem) but I don't know how to drop this assumption. Any help would be much appreciated.

• – Jyrki Lahtonen Nov 22 '17 at 13:02
• Heh, indeed. I just read through his post. I actually have been working with a similar train of thought it seems, though with not much luck. I don't suppose you have any hints? I'm kind of desparate. I've been thinking about this for the past few days. – user433011 Nov 22 '17 at 13:06
• I'm trying to solve the translation of this into group theory. Alas, I need to get back to my main work soon. Mind you, the other poster is also from a province in western Canada. From a city IIRC boasting a rollercoaster inside a mall :-) – Jyrki Lahtonen Nov 22 '17 at 13:10
• Well, I appreciate the effort. I still wonder if this is actually true. The statement seems a bit strong to me, though i'm a beginner with Galois theory. – user433011 Nov 22 '17 at 13:19
• So far I have only found an occasion where the corresponding statement is false for the extension $L/K$ (a carefully chosen $K$). Don't know about $K/F$. – Jyrki Lahtonen Nov 22 '17 at 13:20