Persistent Link:
http://hdl.handle.net/10150/289181
Title:
Galois groups and Greenberg's conjecture
Author:
Marshall, David Clark
Issue Date:
2000
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
We consider the structure of a certain infinite Galois group over Q(ζp) the cyclotomic field of p-th roots of unity. Namely, we consider the Galois group of the maximal p-ramified pro- p-extension. Very little is known about this group. It has a free pro-p presentation in terms of g generators and s relations where g and s may be explicitly computed in terms of the p-rank of the class group of Q(ζp). The structure of the relations in the Galois group are shown to be very closely related to the relations in a certain Iwasawa module. The main result of this dissertation shows this Iwasawa module to be torsion free for a large class of cyclotomic fields. The result is equivalent to verifying Greenberg's pseudo-null conjecture for the given class of fields. As one consequence, we provide a large class of examples of cyclotomic fields which do not admit free pro-p-extensions of maximal rank.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
McCallum, William G.

Full metadata record

DC FieldValue Language
dc.language.isoen_USen_US
dc.titleGalois groups and Greenberg's conjectureen_US
dc.creatorMarshall, David Clarken_US
dc.contributor.authorMarshall, David Clarken_US
dc.date.issued2000en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractWe consider the structure of a certain infinite Galois group over Q(ζp) the cyclotomic field of p-th roots of unity. Namely, we consider the Galois group of the maximal p-ramified pro- p-extension. Very little is known about this group. It has a free pro-p presentation in terms of g generators and s relations where g and s may be explicitly computed in terms of the p-rank of the class group of Q(ζp). The structure of the relations in the Galois group are shown to be very closely related to the relations in a certain Iwasawa module. The main result of this dissertation shows this Iwasawa module to be torsion free for a large class of cyclotomic fields. The result is equivalent to verifying Greenberg's pseudo-null conjecture for the given class of fields. As one consequence, we provide a large class of examples of cyclotomic fields which do not admit free pro-p-extensions of maximal rank.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematics.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorMcCallum, William G.en_US
dc.identifier.proquest9983896en_US
dc.identifier.bibrecord.b40825590en_US
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