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Zaharescu, Alexandru Landquist, Eric 2015-09-28T15:20:11Z 2015-09-28T15:20:11Z 10000-01-01 2009 2009 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 Hasse-Weil Theorem. We then employ Shanks' Baby Step-Giant 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 g-1)/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 S-regulators. 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. Made available in DSpace on 2015-09-28T15:20:11Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3363008.pdf: 3926833 bytes, checksum: 7ed0ec2cbf9e227cdcd4dda8073165c8 (MD5) Previous issue date: 2009 Embargo set by: Seth Robbins for item 88203 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 194 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2009. http://hdl.handle.net/2142/86922 (MiAaPQ)AAI3363008 eng Mathematics Infrastructure, Arithmetic, and Class Number Computations in Purely Cubic Function Fields of Characteristic at Least 5 text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.