Let we have the following tower of extensions: $K \subset L \subset M$, where $L,M$ are Galois over $K$ and $L$ contains all roots of unity of degree $n = \left[M:L\right]$, $Gal(M/L)$ is a cyclic group of order $n$, i.e. $M$ is a Kummer extension.
Is the following claim valid?
If $F \supset K$ is a Galois sub-extension of $M$ then $F \subseteq L$
If yes, could anybody give me a hint how to prove it? If not, could somebody provide me a counter-example?