Identify all the subfields of a field , with elements where is a prime. Draw the lattice of all subfields.
I am allowed to just use a theorem, but I don't know one to use.
The subgroups of are where . Furthermore, if and only if .
Now you can draw the diagram of all subfields.
Hint: The subfield diagram should look like an inverted diagram*.
(*This is not an accident! It follows from fundamental theorem of Galois theory.)