Title: Kronecker's Theory of Binary Bilinear Forms with Applications to Representations of Integers as Sums of Three Squares

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 defense we shall discover the key statements within Kronecker's paper and offer insight into new, detailed arithmetic proofs. Further, I will present some additional results on the proper and complete class numbers for bilinear forms, before demonstrating their use in rigorously developing the connection between binary bilinear forms and binary quadratic forms. 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.