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Transcendence degree in power series rings

Let 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. / Ph. D.

Identiferoai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/37802
Date13 May 2010
CreatorsBoyd, David Watts
ContributorsMathematics, Arnold, Jimmy T., Crofts, G. W., Feustel, C. D., McCoy, Robert A., Sheldon, P. B.
PublisherVirginia Tech
Source SetsVirginia Tech Theses and Dissertation
LanguageEnglish
Detected LanguageEnglish
TypeDissertation, Text
Format25 leave, BTD, application/pdf, application/pdf
RightsIn Copyright, http://rightsstatements.org/vocab/InC/1.0/
RelationOCLC# 22121505, LD5655.V856_1975.B695.pdf

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