Galois theory by calculator

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 $f(x)=x^n+px+q$.

8 pages v2 Corrected typo in author's name