Geometry of Numbers Proof of Götzkyʹs Four-Squares Theorem Original Research Article

The totally positive algebraic integers of certain number fields have been shown to be the sums of four squares of integers from their respective fields. The case of image(image) was demonstrated by Götzky and the cases of image( image) and image( image) were demonstrated by Cohn. In the latter situation, only those integers with even coefficient on the radical term could possibly be represented by sums of squares. These results utilized modular functions in order to get the exact number of representations. Here a method of Grace is adapted to show the existence of a four-squares representation for image( image) without, however, obtaining the number of these. Also, results about the representation of primes by sums of two squares are obtained for image( image).