Advisor:  Dr. Rachel Pries

Committee:  Dr. Jeff Achter, Dr. Mark Shoemaker, Dr. Maria Rugenstein

Title: Relative oriented class groups of quadratic extensions

Abstract: Perhaps the most beautiful development in the theory of binary quadratic forms was the work of Gauss in his Disquisitiones Arithmeticae in which he described the underlying group structure of binary quadratic forms and what we now know as Gauss composition. The power of this discovery lies in a correspondence between binary quadratic forms and ideal class groups of quadratic number fields. In 2004, a series of articles of Bhargava explained geometric approaches to understanding composition laws for binary quadratic forms and generalizations involving correspondences for higher degree number fields. In 2019 Zemková extended the work of Bhargava in the context of relative quadratic extensions and binary quadratic forms of number fields by introducing relative oriented class groups. Zemková explicitly computed these groups for totally real quadratic extensions of the rationals. We extend Zemková’s result by considering extensions L/K with K a totally real quadratic number field and L a relative quadratic extension of K.