Suppose K is a Galois extension of F of degree for some prime p and some . Show there are Galois extensions of F contained in K of degrees and .

So what I want is Fields or

From the fundemental theorem of Galois I get the two diagrams relating the groups to the fields

according to another part of the fundemental theorem I get

and

I get that is Galios over F iff is a normal subgroup of of G.

For some reason I think that G must be cyclic but I can't prove why. If that Is the case I am done.

Thanks