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A survey of results toward the class number problem for real quadratic fields Nevin, Joshua Borwein
Abstract
The class number problem is one of the central open problems of algebraic number theory. It has long been known that there are only finitely many imaginary quadratic fields of class number one, and the full list of such fields is given by the Stark-Heegner theorem, but the corresponding problem for real quadratic fields is still open. It is conjectured that infinitely many real quadratic fields have class number one but at present it is still unknown even whether infinitely many algebraic number fields have class number one. This thesis reviews the relevant work that has been done on this problem in the last several decades. It is primarily concerned with a heuristic model of the sequence of real quadratic class groups called the Cohen-Lenstra heuristics, since this appears to offer the best hope of potentially solving the class number problem. The work of several other people who have put forward interpretations of the Cohen-Lenstra heuristics in other contexts, or who have generalized the heuristics, is also reviewed.
Item Metadata
Title |
A survey of results toward the class number problem for real quadratic fields
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2015
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Description |
The class number problem is one of the central open problems of algebraic number theory. It has long been known that there are only finitely many imaginary quadratic fields of class number one, and the full list of such fields is given by the Stark-Heegner theorem, but the corresponding problem for real quadratic fields is still open. It is conjectured that infinitely many real quadratic fields have class number one but at present it is still unknown even whether infinitely many algebraic number fields have class number one. This thesis reviews the relevant work that has been done on this problem in the last several decades. It is primarily concerned with a heuristic model of the sequence of real quadratic class groups called the Cohen-Lenstra heuristics, since this appears to offer the best hope of potentially solving the class number problem. The work of several other people who have put forward interpretations of the Cohen-Lenstra heuristics in other contexts, or who have generalized the heuristics, is also reviewed.
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Genre | |
Type | |
Language |
eng
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Date Available |
2015-04-20
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution 2.5 Canada
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DOI |
10.14288/1.0166230
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2015-05
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Campus | |
Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution 2.5 Canada