A Universal Quaternary Quadratic Form over Gaussian Integers

Abstract

In this article we show that the form $x^2+i{y}^2+z^2+i{w}^2$ represents all gaussian integers. The main tools used in this proof are Fermat's little theorem (over finite field extensions), the Mordell-Niven theorem (representation of some gaussians), and the generalized Euler-identity over finite field extensions.