An examination of class number for [reproduction of quadratic extensions] where [reproduction of square root of d] has continued fraction expansion of period three
Brent Young · NC Digital Online Collection of Knowledge and Scholarship (The University of North Carolina at Greensboro) · 2009
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.