The first part follows from the fact that for any algebraic elements $a_i$ the following holds:
where $F$ is the field generated by the $a_i$. The second part is giving me trouble. The $a$ has to be such that for every $\lambda_i$ there exist $\mu_j$ such that the following holds (and vice versa since the fields must be the same):
I don't know where to go from here. Any help would be greatly appreciated.