Transcendental degree in power series rings

Let D be an integral domain with quotient field K. Let D[[x]] and K[[x]] be the power series ring over D and K, respectively. In this paper, we show that either (1) K[[x]] and D[[x]] have the same quotient field or (2) the quotient field of K[[x]] has uncountable transcendence degree over that of D[[x]], i.e., tr.d.(K[[x]]/D[[x]])≥ℵ1. In (2), the bound ℵ1 is the greatest lower bound that one can obtain since under the continuum hypothesis the cardinality of the quotient field of K[[x]] is exactly ℵ1 provided that K is countable. We also show that the above result holds when K is replaced by any quotient overring DS of D.