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Fundamental Theorem of Galois Theory

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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

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Solution Summary

Fundamental Theorem of Galois Theory is applied.

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