Proving Cyclic Extension of Finite Galois Group L/F

[PIM]

Homework Statement

Let K be a field, and let K' be an algebraic closure of K. Let sigma be
an automorphism of K' over K, and let F be the fix field of sigma. Let L/F
be any finite extension of F.

Homework Equations

Show that L/F is a finite Galois extension whose
Galois group Gal(L/F) is cyclic.

The Attempt at a Solution

 

[PIM]

I thought about the prime subfiled of F, which is isomorphic to F_p or Q, and tried to prove that L is finite Galois over this prime subfield. (but failed) if I could show this, then it's obvious that L is finite galois over F since F is an intermediate field.

But for the cylic galois group, I still haven't got any idea.