The Structure of the Class Group of Imaginary Quadratic Fields

Let Q( √−d) be an imaginary quadratic field with discriminant Δ. We use the isomorphism between the ideal class groups of the field and the equivalence classes of binary quadratic forms to find the structure of the class group. We determine the structure by combining two of Shanks’ algorithms [7, 8]. We utilize this method to find fields with cyclic factors that have order a large power of 2, or fields with class groups of high 5-ranks or high 7-ranks. To my parents, who never gave up. iii Acknowledgements A massive amount of thanks goes to my advisor, Dr. Charles Parry, for his extensive involvement in generating this work. Without his support, dedication, and limitless supply of patience, this paper would not have been realized. Most of all, I am indebted to him for having confidence in me when I was lacking. I also want to thank the graduate students in McBryde 465 for their constant encouragement an