Determine the Galois group over Q of the splitting field of the question. List all of the subgroups of the Galois group. List all of the subfield of the splitting field of the equation.
(X^7)-1
Hint: .
Find the subgroups of it and correspond them to the intermediate fields.
For instance, if you choose a generator of the above group as , then one of the subgroups of it is .
The reason why I choose the above generator (among them) is because 3 has order 6 in ( Meanwhile, 2 has order 3 in ).
Since , the corresponding intermediate field is .
I'll leave it to you to find the remaining subgroups of and their corresponding intermediate fields.