# Gaussian Integers, Rings, Finite Fields, and the Magic Square of Squares

## TL;DR

This paper explores the problem of constructing a 3x3 magic square of squares through algebraic and number theoretic methods, including Gaussian integers, finite fields, and rings, providing new insights and computational tools.

## Contribution

It introduces a novel approach linking the magic square problem to quartic polynomials over abelian extensions and develops a new search method using Gaussian integers.

## Key findings

- Equivalent formulation of the problem using quartic polynomials
- A new search method based on Gaussian integers
- Conjectures on rings and finite fields supporting magic squares

## Abstract

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.