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.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

1975

Author dc:creator

• Boyd, David Watts

Chair dc:contributor.committeechair

• Arnold, Jimmy T.

Committee members dc:contributor.committeemember

• Crofts, G. W.
• Feustel, C. D.
• McCoy, Robert A.
• Sheldon, P. B.

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Dc Identifier Other

etd-05132010-132826

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/37802