The set of non-zero elements of a finite field of order q form a cyclic group of order q−1 under multiplication. If q>2 then this cyclic group has at least one non-identity element α that generates it.
Choose α∈F16 that generates all the 15 non-zero elements.
Then α5 generates the cyclic subgroup of order 3 corresponding to the subfield isomorphic to F4.
α3 generates the subgroup of order 5, but 5+1=6 is not a power of 2, so there is no corresponding subfield.
α15=1 generates the subgroup of order 1, corresponding to the subfield isomorphic to F2.
In general, Fpn has subfields isomorphic to Fpk where k is a divisor of n.
