Mathematics faculty and graduate students are invited to attend the PhD Defense of Kelly O’Connor.

Also available on Zoom:  https://zoom.us/j/97617876065?pwd=OWdsWVNIVnVEN1RMdXV4ZEZYZXRvUT09

Meeting ID: 976 1787 6065

Passcode: 477238

Advisor:  Dr. Rachel Pries

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

Title:  Relative oriented class groups of quadratic extensions

Abstract:   In 2019 Zemková defined relative oriented class groups associated to quadratic extensions of number fields L/K, extending work of Bhargava concerning composition laws for binary quadratic forms over number fields of higher degree. This work generalized the classical correspondence between ideal class groups of quadratic number fields and integral binary quadratic forms to any base number field of narrow class number one. Zemková explicitly computed these relative oriented class groups for quadratic extensions of the rationals. We consider extended versions of this work and develop general strategies to compute relative oriented class groups for quadratic extensions of totally real number fields by way of the action of the Galois group of K/Q on the set of real embeddings of K. We also investigate the binary quadratic forms side of Zemkova’s bijection and determine conditions for representability of elements of K by binary quadratic forms defined over K.