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Class numbers of totally real number fields
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TitleInfo
Title
Class numbers of totally real number fields
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Miller
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John C.
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John C. Miller
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Iwaniec
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Henryk
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Henryk Iwaniec
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chair
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Kontorovich
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Alex
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Alex Kontorovich
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internal member
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Tunnell
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Jerrold
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Washington
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Lawrence
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Lawrence Washington
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Rutgers University
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Graduate School - New Brunswick
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theses
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2015
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2015-05
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2015
Place
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xx
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eng
Abstract
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The determination of the class number of totally real fields of large discriminant is known to be a difficult problem. The Minkowski bound is too large to be useful, and the root discriminant of the field can be too large to be treated by Odlyzko's discriminant bounds. This thesis describes a new approach. By finding nontrivial lower bounds for sums over prime ideals of the Hilbert class field, we establish upper bounds for class numbers of fields of larger discriminant. This analytic upper bound, together with algebraic arguments concerning the divisibility properties of class numbers, allows us to determine the class numbers of many number fields that have previously been untreatable by any known method. For example, we consider the cyclotomic fields and their real subfields. Surprisingly, the class numbers of cyclotomic fields have only been determined for fields of small conductor, e.g. for prime conductors up to 67, due to the problem of finding the class number of its maximal real subfield, a problem first considered by Kummer. Our results have significantly improved the situation. We also study the cyclotomic Z_p-extensions of the rationals. Based on the heuristics of Cohen and Lenstra, and refined by new results on class numbers of particular fields, we provide evidence for the following conjecture first suggested by Coates: For all primes p, every number field in a cyclotomic Z_p-extension of Q has class number 1.
Subject
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Topic
Mathematics
Subject
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Topic
Cyclotomy
Subject
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Topic
Algebraic fields
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Rutgers University Electronic Theses and Dissertations
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ETD_6332
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electronic resource
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1 online resource (v, 105 p.)
Note
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Ph.D.
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Includes bibliographical references
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by John C. Miller
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Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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Identifier
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doi:10.7282/T3474CQ7
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ETD doctoral
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Rights
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The author owns the copyright to this work.
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Name
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Miller
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John
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C.
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2015-04-13 12:21:06
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John Miller
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Rutgers University. Graduate School - New Brunswick
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Author Agreement License
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
Copyright
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Copyright protected
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Open
Reason
Permission or license
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windows xp
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