(With S, Akhtari and M. Widmer) Let K be an algebraic number field and H the absolute Weil
height. Write c_K for a certain positive constant that is an invariant of K.
We consider the question: does K contain an algebraic integer alpha such that
both K = Q(alpha ) and H(alpha)≤ c_K? If K has a real embedding then a positive
answer was established in previous work. Here we obtain a positive answer
if Tor(K^x) ≠ {\pm 1}, and so K has only complex embeddings. We also show
that if the answer is negative, then K is totally complex, Tor(K^x) = {\pm 1},
and K is a Galois extension of its maximal totally real sub field F. Thus if the
answer is negative then K looks much like a CM- field, and we conjecture that
K is a CM- field.

If time permits we will also describe a new equidistribution theorem for a
special multiplicative subgroup that occurs in a CM- field.