Let . Compute the degree of over and find a basis.

Note that is a root of , which is irreducible over by Eisenstein and thus is the minimal polynomial of over . So has degree 2 over , with a basis. Suppose there exist such that ; comparing coefficients, we see that . However, the left hand side of this equation is positive, while the right hand side is negative- a contradiction. In particular, is irreducible over , and hence is the minimal polynomial of over . So has degree 4. We may take as a basis the set .

Note that the minimal polynomial of over is . (Use Eisenstein to show irreducibility.) By this previous exercise, is also irreducible over . Thus has degree 12 over . We may take as a basis the set .