Boyd, David Watts2014-03-142014-03-141975-05-09etd-05132010-132826http://hdl.handle.net/10919/37802Let D[[X]] be the ring of formal power series over the commutative integral domain D. Gilmer has shown that if K is the quotient field of D, then D[[X]] and K[[X]] have the same quotient field if and only if K[[X]] ~ D[[X]]D_(O). Further, if a is any nonzero element of D, Sheldon has shown that either D[l/a][[X]] and D[[X]] have the same quotient field, or the quotient field of D[l/a][[X]] has infinite transcendence degree over the quotient field of D[[X]]. In this paper, the relationship between D[[X]] and J[[X]] is investigated for an arbitrary overring J of D. If D is integrally closed, it is shown that either J[[X]] and D[[X]] have the same quotient field, or the quotient field of J[[X]] has infinite transcendence degree over the quotient field of D[[X]]. It is shown further, that D is completely integrally closed if and only if the quotient field of J[[X]] has infinite transcendence degree over the quotient field of D[[X]] for each proper overring J of D. Several related results are given; for example, if D is Noetherian, and if J is a finite ring extension of D, then either J[[X]] and D[[X]] have the same quotient field or the quotient field of J[[X]] has infinite transcendence degree over the quotient field of D[[X]]. An example is given to show that if D is not integrally closed, J[[X]] may be algebraic over D[[X]] while J[[X]] and ~[[X]] have dif~erent quotient fields.25 leaveBTDapplication/pdfenIn CopyrightLD5655.V856 1975.B695Power series ringsDissertationhttp://scholar.lib.vt.edu/theses/available/etd-05132010-132826/