Binary quadratic forms and genus theory

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Rick L. Shepherd (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site:
Brett Tangedal

Abstract: The study of binary quadratic forms arose as a natural generalization of questions about the integers posed by the ancient Greeks. A major milestone of understanding occurred with the publication of Gauss's Disquisitiones Arithmeticae in 1801 in which Gauss systematically treated known results of his predecessors and vastly increased knowledge of this part of number theory. In effect, he showed how collections of sets of binary quadratic forms can be viewed as groups, at a time before group theory formally existed. Beyond that, he even defined and calculated genus groups, which are essentially quotient groups, that explain which congruence classes of numbers can be represented by given sets of forms. This thesis examines Gauss's main results as interpreted and refined over two centuries. Also code has been created to implement many of the algorithms used in studying the relationships of such forms to each other, to generate examples, and to provide a small toolkit of software for analyzing the corresponding algebraic structures.

Additional Information

Language: English
Date: 2013
Binary quadratic forms, Class groups, Class number, Genus theory, PARI code
Forms, Binary
Forms, Quadratic
Group theory

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