Title page for ETD etd-05132010-132826


Type of Document Dissertation
Author Boyd, David Watts
URN etd-05132010-132826
Title Transcendence degree in power series rings
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Arnold, J. T. Committee Chair
Crofts, G. W. Committee Member
Feustel, C. D. Committee Member
McCoy, Robert A. Committee Member
Sheldon, P. B. Committee Member
Keywords
  • Power series rings
Date of Defense 1975-05-09
Availability restricted
Abstract
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.
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