Is the elements in GF((2^2)^2) same as the elements in GF(2^4) ? and why is that so?
The notation GF(2^4) is already an ambiguous notation because it does not refer to any specific field. It happens to be that all fields of 16 elements are isomorphic to eachother and so we say "the" field of 16 elements if GF(2^4) however the elements may be totally distinct.