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2026
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| Online Access: | https://doi.org/10.5281/zenodo.19903519 |
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| _version_ | 1866901180989308928 |
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| author | ueda, takashi |
| author_facet | ueda, takashi |
| contents | <p>We establish a complete criterion for the solvability by radicals of</p> <p>algebraic equations in standard form, where non-zero terms are spec-</p> <p>ified by a set of indices M ⊆ {1, 2, . . . , n} with arbitrary complex</p> <p>coefficients. The criterion is based on the residue classes of indices</p> <p>modulo 4, termed channels. We prove that an equation in standard</p> <p>form is solvable by radicals if and only if it satisfies the channel pri-</p> <p>ority rule: when indices from multiple channels are present, only the</p> <p>smallest index (i.e., the term of highest degree) may be retained from</p> <p>each channel; when only a single channel is used, any index within that</p> <p>channel is permitted. The proof reduces the problem to equations of</p> <p>degree at most 4, which are classically solvable.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19903519 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Channel Priority and Algebraic Solvability: A Complete Characterisation of Solvable Equations in Standard Form ueda, takashi <p>We establish a complete criterion for the solvability by radicals of</p> <p>algebraic equations in standard form, where non-zero terms are spec-</p> <p>ified by a set of indices M ⊆ {1, 2, . . . , n} with arbitrary complex</p> <p>coefficients. The criterion is based on the residue classes of indices</p> <p>modulo 4, termed channels. We prove that an equation in standard</p> <p>form is solvable by radicals if and only if it satisfies the channel pri-</p> <p>ority rule: when indices from multiple channels are present, only the</p> <p>smallest index (i.e., the term of highest degree) may be retained from</p> <p>each channel; when only a single channel is used, any index within that</p> <p>channel is permitted. The proof reduces the problem to equations of</p> <p>degree at most 4, which are classically solvable.</p> |
| title | Channel Priority and Algebraic Solvability: A Complete Characterisation of Solvable Equations in Standard Form |
| url | https://doi.org/10.5281/zenodo.19903519 |