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Title:  Infrastructure, Arithmetic, and Class Number Computations in Purely Cubic Function Fields of Characteristic at Least 5 
Author(s):  Landquist, Eric 
Doctoral Committee Chair(s):  Zaharescu, Alexandru 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  Finally, we describe methods to compute the divisor class number, h, of K, and in the case that O has unit rank 1 or 2, the regulator and ideal class number of O as well. A method of Scheidler and Stein [SS07, SS08] determines sharper upper and lower bounds on h, for a given cubic function field, than those given by the HasseWeil Theorem. We then employ Shanks' Baby StepGiant Step algorithm [Sha71] and Pollard's Kangaroo method [Pol78], to search this interval and compute the desired invariants for purely cubic function fields of unit rank 0 and 1. The total complexity of the method to compute these invariants is O (q(2 g1)/5+epsilon(g )) ideal operations as q → infinity, where 0 ≤ epsilon( g) ≤ 1/5. With this approach, we computed the 28 decimal digit divisor class numbers of two purely cubic function fields of genus 3: one of unit rank 0 and one of unit rank 1. We also computed the 25 decimal digit divisor class numbers of two purely cubic function fields of genus 4: one of unit rank 0 and one of unit rank 1. In the unit rank 1 examples, we factored the divisor class numbers into the ideal class numbers and the respective 26 and 24 decimal digit Sregulators. We believe that these are the largest divisor class numbers ever computed for a cubic function field of genus at least 4 and the largest regulators ever computed for any cubic function field, respectively. 
Issue Date:  2009 
Type:  Text 
Language:  English 
Description:  194 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 2009. 
URI:  http://hdl.handle.net/2142/86922 
Other Identifier(s):  (MiAaPQ)AAI3363008 
Date Available in IDEALS:  20150928 
Date Deposited:  2009 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois