LAUSR

Abstract 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., t r . d . ( K [ [ x ] ] / D [ [ x ] ] ) ≥ ℵ 1 . In (2), the bound ℵ 1 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 ℵ 1 provided that K is countable. We also show that the above result holds when K is replaced by any quotient overring D S of D.