  To what extent does the maximal subfield spectrum of a division algebra
determine the isomorphism class of that algebra? It has been shown that over
some fields a quaternion division algebra's isomorphism class is largely if not
entirely determined by its maximal subfield spectrum. However in this paper, we
show that there are fields for which the maximal subfield spectrum says little
to nothing about a quaternion division algebra's isomorphism class. We give an
explicit construction of a division algebra with infinite genus. Along the way
we introduce the notion of a "linking field extension," which we hope will be
of independent interest. We go on to show that there exists a field K for which
(1) there are infinitely many nonisomorphic quaternion division algebras with
center K, and (2) any two quaternion division algebra with center K are
pairwise weakly isomorphic. In fact we show that there are infinitely many
nonisomorphic fields satisfying these two conditions.
