University of Vermont ScholarWorks @ UVM Graduate College Dissertations and Theses Dissertations and Theses 2017 Hilbert Class Fields of Imaginary Quadratic Fields and Reflex ieldF s of Certain Sextic CM Fields Garvin Gaston University of Vermont Follow this and additional works at: https://scholarworks.uvm.edu/graddis Part of the Mathematics Commons Recommended Citation Gaston, Garvin, "Hilbert Class Fields of Imaginary Quadratic Fields and Reflex ieF lds of Certain Sextic CM Fields" (2017). Graduate College Dissertations and Theses. 808. https://scholarworks.uvm.edu/graddis/808 This Thesis is brought to you for free and open access by the Dissertations and Theses at ScholarWorks @ UVM. It has been accepted for inclusion in Graduate College Dissertations and Theses by an authorized administrator of ScholarWorks @ UVM. For more information, please contact
[email protected]. Hilbert Class Fields of Imaginary Quadratic Fields and Reflex Fields of Certain Sextic CM Fields A Thesis Presented by Garvin Gaston to The Faculty of the Graduate College of The University of Vermont In Partial Fulfillment of the Requirements for the Degree of Master of Science Specializing in Mathematics October, 2017 Defense Date: August 2, 2017 Thesis Examination Committee: Christelle Vincent, Ph.D., Advisor Christian Skalka, Ph.D., Chairperson Richard Foote, Ph.D. Cynthia J. Forehand, Ph.D., Dean of Graduate College Abstract In this thesis we look at particular details of class field theory for complex multipli- cation fields. We begin by giving some background on fields, abelian varieties, and complex multiplication. We then turn to the first task of this thesis and give an implementation in Sage of a classical algorithm to compute the Hilbert class field of a quadratic complex multiplication field using the j-invariant of elliptic curves with complex multiplication by the ring of integers of the field, and we include three explicit examples to illustrate the algorithm.