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 Author Constable, Jonathan A. Title Kronecker's Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares URL https://uknowledge.uky.edu/math_etds/35 https://uknowledge.uky.edu/cgi/viewcontent.cgi?article=1034&context=math_etds Publication Date 2016 Degree Level doctoral University/Publisher University of Kentucky Abstract 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 Country of Publication us Format application/pdf Record ID oai:uknowledge.uky.edu:math_etds-1034 Repository uky-diss Date Retrieved 2019-06-25 Date Indexed 2019-06-26

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…Churchill The theory of binary quadratic forms is a source of classical problems in number theory and has been studied extensively. A lesser known paper by Leopold Kronecker in 1883 [Kr1897] contains a novel manner for connecting the classical…

…class number theory of binary quadratic forms to the class number for binary bilinear forms. Although correct, Kronecker’s paper requires prior knowledge of several key results in order to construct his ultimate result; a purely arithmetic proof of the…

…of the classical treatment for automorphs of binary quadratic forms as given by Flath in [Fl1989, p. 125]. Next, Chapter 3 commences our journey towards understanding Kronecker’s paper [Kr1897]. We begin with Kronecker’s definition…

…x28;A12 + A21 ) xy + A22 y 2 . Lemma 2.4.9. The binary quadratic forms ax2 + 2bxy + cy 2 , a, b, c ∈ Z are a subset of the bilinear forms with integer coefficients. Proof. Let ax2 +2bxy +cy 2 where a, b, c ∈ Z be a binary quadratic form. Consider…

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