Galois theory by calculator

2025/08/26 by Thomas W. Mattman, Mattman, Thomas W., Dylan Robertson-Figaniak +3 · 1 voice
Mathematics · #12F10 #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.2508.18595

arxiv published 2025/08/26 · arxiv updated 2025/08/27

Abstract

We present an algorithm to determine the Galois group of an irreducible monic polynomial f(x) ∈ 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) = xⁿ + px + q.

Discussions

• Galois Theory by Calculator [hn, 2 points, 1 comments]

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