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arXiv · 2508.18595

Galois theory by calculator

Thomas W. Mattman·Dylan Robertson-Figaniak·Zoe Steele
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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 $f(x) = x^n + px + q$.

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Thomas W. Mattman, Dylan Robertson-Figaniak, Zoe Steele. 2025-08-27. Galois theory by calculator. https://arxiv.org/abs/2508.18595

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