Higher Composition Laws

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Rezazadeh, Sina | 2012

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 43733 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Pournaki, Mohammad Reza; Shahshahani, Siavash
Abstract:
Gauss' theory of the arithmetic of quadratic forms appeared in Disquisitiones Arithmeticae (1801) and in particular Gauss presented a composition law for binary quadratic forms and related it to the arithmetic of quadratic extensions of Q. In the series of papers of “Higher Composition Laws” Manjul Bhargava gave a far reaching generalization of Gauss' composition law and extended it to binary cubic, and a number of other cases. He obtained six composition laws one of which the classical one of Gauss. Since the work of Gauss has had deep impact on number theory and in particular on the arithmetic of quadratic fields, one expects that Bhargava’s theory to lead to new insights in the field. In this thesis we study the main idea of higher composition laws of Bhargava and their applications.
Keywords:
Quadratic Forms ; Gauss Composition Law ; Higher Composition Laws ; Class Group ; Bhargava's Cube

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