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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.18595 |
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| _version_ | 1866911124417413120 |
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| author | Mattman, Thomas W. Robertson-Figaniak, Dylan Steele, Zoe |
| author_facet | Mattman, Thomas W. Robertson-Figaniak, Dylan Steele, Zoe |
| contents | 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 $f(x) = x^n + px + q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18595 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Galois theory by calculator Mattman, Thomas W. Robertson-Figaniak, Dylan Steele, Zoe Number Theory 12F10 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 $f(x) = x^n + px + q$. |
| title | Galois theory by calculator |
| topic | Number Theory 12F10 |
| url | https://arxiv.org/abs/2508.18595 |