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Constable, Jonathan A.
Kronecker's Theory of *Binary* *Bilinear* *Forms* with Applications to Representations of Integers as Sums of Three Squares.

Degree: 2016, University of Kentucky

URL: https://uknowledge.uky.edu/math_etds/35

In 1883 Leopold Kronecker published a paper containing “a few explanatory remarks” to an earlier paper of his from 1866. His work loosely connected the theory of integral binary bilinear forms to the theory of integral binary quadratic forms. In this dissertation we discover the statements within Kronecker's paper and offer detailed arithmetic proofs. We begin by developing the theory of binary bilinear forms and their automorphs, providing a classification of integral binary bilinear forms up to equivalence, proper equivalence and complete equivalence.
In the second chapter we introduce the class number, proper class number and complete class number as well as two refinements, which facilitate the development of a connection with binary quadratic forms.
Our third chapter is devoted to deriving several class number formulas in terms of divisors of the determinant. This chapter also contains lower bounds on the class number for bilinear forms and classifies when these bounds are attained.
Lastly, we use the class number formulas to rigorously develop Kronecker's connection between binary bilinear forms and binary quadratic forms. We supply purely arithmetic proofs of five results stated but not proven in the original paper. We conclude by giving an application of this material to the number of representations of an integer as a sum of three squares and show the resulting formula is equivalent to the well-known result due to Gauss.

Subjects/Keywords: complete equivalence; binary bilinear forms; binary quadratic forms; class number relations; L. Kronecker; Gauss; Algebra; Number Theory

…Chapter 5 A Connection between *Bilinear* *Forms* and *Binary* Quadratic *Forms*
5.1 Developing the… …number for *binary* *bilinear* *forms*. Although correct, Kronecker’s paper requires prior knowledge… …the theory of automorphs for
*binary* *bilinear* *forms*. Our treatment follows in the manner of… …Kronecker’s connection
between *binary* quadratic *forms* and *binary* *bilinear* *forms*. We supply proofs of… …define three types of class number for positive definite *binary* *bilinear* *forms*.
Definition…

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APA (6^{th} Edition):

Constable, J. A. (2016). Kronecker's Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares. (Doctoral Dissertation). University of Kentucky. Retrieved from https://uknowledge.uky.edu/math_etds/35

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

Constable, Jonathan A. “Kronecker's Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares.” 2016. Doctoral Dissertation, University of Kentucky. Accessed July 15, 2020. https://uknowledge.uky.edu/math_etds/35.

MLA Handbook (7^{th} Edition):

Constable, Jonathan A. “Kronecker's Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares.” 2016. Web. 15 Jul 2020.

Vancouver:

Constable JA. Kronecker's Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares. [Internet] [Doctoral dissertation]. University of Kentucky; 2016. [cited 2020 Jul 15]. Available from: https://uknowledge.uky.edu/math_etds/35.

Council of Science Editors:

Constable JA. Kronecker's Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares. [Doctoral Dissertation]. University of Kentucky; 2016. Available from: https://uknowledge.uky.edu/math_etds/35