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?
Ok, I think I got it now:
So, I say that
And since V has 3 subgroups of order 2, which are and , then there are 3 such that and those are and since the splitting field for is .
And then I just have to check that those 3 are independent from each other and thereby I have found the 3 's
Did I get that right?