Assignment:
Suppose K is a Galois extension of F of degree p^n for some prime p and some n >= 1. Show that there are Galois extensions of F contained in K of degree p and p^(n-1).
Note: It is assumed that you know the following results in order to prove this theorem.
You may search for the proofs of these results.
1. G = Aut_F(K) has a subgroup of order p^(n-1)
2. The Fundamental Theorem of Galois Theory
3. Cauchy's Theorem
Provide complete and step by step solution for the question and show calculations and use formulas.