  We show the 3 by 3 magic square of squares problem equivalent to solving
quartic polynomials with certain factorization constraints over an abelian
extension of the rationals. We analyze a particular case in which said
extension is assumed to be the Gaussian integers resulting a new search method.
Additionally, the magic square of squares is analyzed over finite fields and
rings of the form Z/nZ resulting in some conjectures enumerating the rings and
finite fields in which a magic square of squares can be constructed. Code is
made available.
