In this paper we
determine the Q(i)-division rings which have maximal subfields of the form E(i),
where E∕Q is cyclic and i =. These are precisely the Q(i)-division
rings having maximal subfields which are abelian over Q. More generally
we determine the Q(i)-division rings having maximal subfields which are
Galois over Q. We show that a division ring D contains such subfields if
and only if the same is true for the 2-part of the Sylow decomposition of
D.
. These are precisely the 