Hi everybody. I have this exercise where I am to describe all the subfields and of and with and , where and are the splitting fields for the polynomials and over .

(Remark: is the cyclotomic field )

Now, I would suppose that you have to use the transitivity theorem to say that;

and then argue that or did I get that wrong? I'm just not sure, then, how to argue that I then have found ALL the subfields with this dimension?