Integers of quadratic fields as sums of squares

Abstract

m being a positive square-free rational integer, is expressible as a sum of three squares of integers of the field. Gaussian integers are treated in ?3, integers of the general imaginary quadratic field in ?4; necessary and sufficient conditions for two-square sums are given in each case. Section 6 treats real quadratic integers, and ?7 interprets some of the results in the theory of Diophantine equations. It will be recalled that the coefficients of quadratic integers are not always rational integers. Specifically, if the field is an extension of the rational number field by 0 in equation (1), and if m-3 (mod 4), the integers of the field are given by

Locations

• Transactions of the American Mathematical Society - View - PDF