It is usually a difficult problem to characterize precisely which elements of a given integral domain can be written as a sum of squares of elements from the integral domain. Let R denote the ring of integers in a quadratic number field. This talk will deal with the problem of identifying which elements of R can be written as a sum of squares. If an element in R can be written as a sum of squares, then the element must be totally positive. This necessary condition is not always sufficient. We will determine exactly when this necessary condition is sufficient. In addition, we will develop several criteria to guarantee that a representation as a sum of squares is possible. The results are based on theorems of I. Niven and C. Siegel from the 1940's, and R. Scharlau from 1980.
