Galois theory by calculator

TL;DR

This paper introduces an accessible algorithm for computing the Galois group of certain polynomials using simple calculator-based checks, making Galois theory more approachable for educational and practical purposes.

Contribution

It provides a novel, calculator-friendly algorithm for determining Galois groups of degree up to five polynomials, simplifying complex algebraic computations.

Findings

Implemented at Desmos.com for practical use.

Simplified algorithm for specific polynomial forms.

Demonstrated feasibility of calculator-based Galois group determination.

Abstract

We present an algorithm to determine the Galois group of an irreducible monic polynomial $f(x) \in \mathbb{Z}[x]$ of degree at most five. Following work of Conrad, Dummit, and Stauduhar this comes down to answering two questions: Is a given integer a square? and Does a given polynomial have an integral root? Since these are both easily addressed with a calculator, our algorithm amounts to Galois theory by calculator. For example, we have an implementation at Desmos.com. In an appendix we present a simplified version of our algorithm, suitable for a handheld calculator, in case $f(x) = x^n + px + q$.