Show that a sequence of fields exists

Question

I do not have a clue how to solve the following problem:

Let $K\subseteq L$ be Galois extension of degree $p^n$, where $p$ is prime and $n$ is natural. Show that there exists a sequence of subfields $K=K_0 \subseteq K_1 \subseteq K_2 \subseteq \ldots \subseteq K_n \subseteq L$, such that $(K_i : K_{i-1}) = p$, where $i=1, \ldots, n$. Moreover, show $K \subseteq K_i$ is Galois extension.

I will be grateful for any help.

Thanks

Answer 1

Every $p$ group has a non trivial center, this gives a normal subgroup of order $p$. It corresponds to your $K_n$. And $K_n$ is then Galois, now just continue by induction.