### Title page for ETD etd-05112005-124308

Type of Document Master's Thesis
Author Miller, Nicole Renee
Author's Email Address nimille2@vt.edu
URN etd-05112005-124308
Title The Structure of the Class Group of Imaginary Quadratic Fields
Degree Master of Science
Department Mathematics
Advisory Committee
Advisor Name Title
Parry, Charles J. Committee Chair
Brown, Ezra A. Committee Member
Haskell, Peter E. Committee Member
Keywords
• 7-rank
• 5-rank
• Positive Definite Forms
• Genera
• Class Group
• Binary Quadratic Fields
Date of Defense 2005-05-11
Availability unrestricted
Abstract
Let $Q(\sqrt{-d})$ be an imaginary quadratic field with

discriminant $\Delta$. 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.

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