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1.
NC DOCKS at The University of North Carolina at Greensboro; Shepherd, Rick L.
*Binary**quadratic* *forms* and genus theory.

Degree: 2013, NC Docks

URL: http://libres.uncg.edu/ir/uncg/f/Shepherd_uncg_0154M_11099.pdf

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.

Subjects/Keywords: Forms, Binary; Forms, Quadratic; Group theory

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

NC DOCKS at The University of North Carolina at Greensboro; Shepherd, R. L. (2013). Binary quadratic forms and genus theory. (Thesis). NC Docks. Retrieved from http://libres.uncg.edu/ir/uncg/f/Shepherd_uncg_0154M_11099.pdf

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

NC DOCKS at The University of North Carolina at Greensboro; Shepherd, Rick L. “Binary quadratic forms and genus theory.” 2013. Thesis, NC Docks. Accessed July 15, 2020. http://libres.uncg.edu/ir/uncg/f/Shepherd_uncg_0154M_11099.pdf.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

NC DOCKS at The University of North Carolina at Greensboro; Shepherd, Rick L. “Binary quadratic forms and genus theory.” 2013. Web. 15 Jul 2020.

Vancouver:

NC DOCKS at The University of North Carolina at Greensboro; Shepherd RL. Binary quadratic forms and genus theory. [Internet] [Thesis]. NC Docks; 2013. [cited 2020 Jul 15]. Available from: http://libres.uncg.edu/ir/uncg/f/Shepherd_uncg_0154M_11099.pdf.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

NC DOCKS at The University of North Carolina at Greensboro; Shepherd RL. Binary quadratic forms and genus theory. [Thesis]. NC Docks; 2013. Available from: http://libres.uncg.edu/ir/uncg/f/Shepherd_uncg_0154M_11099.pdf

Not specified: Masters Thesis or Doctoral Dissertation

University of Toronto

2.
Dahl, Alexander Oswald.
On Moments of Class Numbers of Real *Quadratic* Fields.

Degree: 2010, University of Toronto

URL: http://hdl.handle.net/1807/24553

Class numbers of algebraic number fields are central invariants. Once the underlying field has an infinite unit group they behave very irregularly due to a non-trivial regulator. This phenomenon occurs already in the simplest case of real quadratic number fields of which very little is known. Hooley derived a conjectural formula for the average of class numbers of real quadratic fields. In this thesis we extend his methods to obtain conjectural formulae and bounds for any moment, i.e., the average of an arbitrary real power of class numbers. Our formulae and bounds are based on similar (quite reasonable) assumptions of Hooley's work. In the final chapter we consider the case of the -1 power from a numerical point of view and develop an efficient algorithm to compute the average for the -1 class number power without computing class numbers.

MAST

Subjects/Keywords: analytic number theory; real quadratic fields; binary quadratic forms; class group moments; 0405

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Dahl, A. O. (2010). On Moments of Class Numbers of Real Quadratic Fields. (Masters Thesis). University of Toronto. Retrieved from http://hdl.handle.net/1807/24553

Chicago Manual of Style (16^{th} Edition):

Dahl, Alexander Oswald. “On Moments of Class Numbers of Real Quadratic Fields.” 2010. Masters Thesis, University of Toronto. Accessed July 15, 2020. http://hdl.handle.net/1807/24553.

MLA Handbook (7^{th} Edition):

Dahl, Alexander Oswald. “On Moments of Class Numbers of Real Quadratic Fields.” 2010. Web. 15 Jul 2020.

Vancouver:

Dahl AO. On Moments of Class Numbers of Real Quadratic Fields. [Internet] [Masters thesis]. University of Toronto; 2010. [cited 2020 Jul 15]. Available from: http://hdl.handle.net/1807/24553.

Council of Science Editors:

Dahl AO. On Moments of Class Numbers of Real Quadratic Fields. [Masters Thesis]. University of Toronto; 2010. Available from: http://hdl.handle.net/1807/24553