Contemporary Mathematics A New Look at Sums of Squares Summary: Contemporary Mathematics A New Look at Sums of Squares Larry J. Gerstein Let R be an integral domain. Among the most basic problems in the theory of quadratic forms over R is the determination of which nonzero elements in R can be expressed as a sum of n squares in R, where n is a positive integer. When R = Z, the classics of this genre are the two-square, four-square, and three-square theorems of Fermat, Lagrange, and Gauss, respectively (listed here in chronological order). Of more recent vintage (1964) is this theorem of Cassels: If k is a field of characteristic not 2, then a polynomial in k[x] is a sum of n squares of polynomials if and only if it is a sum of n squares of rational functions in k(x). (In slightly more general form this result is known as the Cassels­Pfister theorem, a cornerstone of the algebraic theory of quadratic forms.) My first goal in this paper is to present a unified approach to all these results. Having done this, I will begin the process of extending this approach to a wider class of problems on representations by quadratic forms. At the end of the paper I will raise some questions that arise naturally along the way. For a ring R, the symbol n will denote the set of nonzero elements in R that can be written as a sum of n squares of elements of R. From now on we will assume R is a principal ideal domain whose quotient field Collections: Mathematics