On unsolvable equations of prime degree
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866918186549510144 |
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| author | Brzeziński, Juliusz Stevens, Jan |
| author_facet | Brzeziński, Juliusz Stevens, Jan |
| contents | Kronecker observed that either all roots or only one root of a solvable irreducible equation of odd prime degree with integer coefficients are real. This gives a possibility to construct specific examples of equations not solvable by radicals. A relatively elementary proof without using the full power of Galois theory is due to Weber. We give a rather short proof of Kronecker's theorem with a slightly different argument from Weber's. Several modern presentations of Weber's proof contain inaccuracies, which can be traced back to an error in the original proof. We discuss this error and how it can be corrected. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14221 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On unsolvable equations of prime degree Brzeziński, Juliusz Stevens, Jan Number Theory History and Overview Primary 12F10, Secondary 12-03 01A55 Kronecker observed that either all roots or only one root of a solvable irreducible equation of odd prime degree with integer coefficients are real. This gives a possibility to construct specific examples of equations not solvable by radicals. A relatively elementary proof without using the full power of Galois theory is due to Weber. We give a rather short proof of Kronecker's theorem with a slightly different argument from Weber's. Several modern presentations of Weber's proof contain inaccuracies, which can be traced back to an error in the original proof. We discuss this error and how it can be corrected. |
| title | On unsolvable equations of prime degree |
| topic | Number Theory History and Overview Primary 12F10, Secondary 12-03 01A55 |
| url | https://arxiv.org/abs/2406.14221 |