An examination of class number for [reproduction of quadratic extensions] where [reproduction of square root of d] has continued fraction expansion of period three

UNCW Author/Contributor (non-UNCW co-authors, if there are any, appear on document)
Brent O. J. Young (Creator)
The University of North Carolina Wilmington (UNCW )
Web Site:
Michael Freeze

Abstract: We begin by ¯nding an appropriate parametrization for d that will ensure pd has continued fraction expansion of period three. After ¯nding this parametrization, we use elementary arguments to limit the possible values of d leading to real quadratic number ¯elds Q(pd) of class number one to a set of 30 congruence classes modulo 9240. We then examine the class number formula, and using an analytic result we are able to place an upper bound on d beyond which the class number must exceed one (with at most one exception). We conclude by examining algorithms that enable us to compute the class num- ber more e±ciently. Using such algorithms (as programmed in PARI) we give an enumeration of all such ¯elds with class number one.

Additional Information

A Thesis Submitted to University North Carolina Wilmington in Partial Ful¯llment Of the Requirements for the Degree of Master of Science
Language: English
Date: 2009
Field extensions (Mathematics), Mathematical optimization
Field extensions (Mathematics)
Mathematical optimization