A Universal Quaternary Quadratic Form over Gaussian Integers

In this article we show that the form $x^2 + iy^2 + z^2 + iw^2$ 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.