Gauss Composition and Bhargava's Cube Law

A basic question in mathematics asks "Which primes are of the form x²+ y²where x and y are integers?" The answer was known to Fermat in 1640 and is easy to state: a prime p is a sum of two squares if and only if p = 2 or p = 1mod4. To generalize this basic question, one might ask "Which primes are of the form x²+ ny²where x, n, and y are integers?" It turns out that this question is much more difficult to answer and yet the solution is still easy to state. The study of x²+ ny²naturally leads to the study of quadratic forms, which are expressions of the form f(x,y) = ax²+ bxy + cy², a, b, c eZ. In 1801, a group law on equivalence classes of quadratic forms was published by Gauss in which the class of x²+ ny² is the identity element. In his 2001 Princeton PhD thesis, Manjul Bhargava developed a series of far-reaching generalizations of Gauss composition that give new information about class groups in number fields of degree < 6. The first of these generalizations is a composition law based on geometric cubes with an integer assigned to each corner. This thesis outlines Gauss composition and introduces the first generalization developed by Bhargava. Using this generalization, a theory of composition of binary cubic forms is introduced.