A degree problem for two algebraic numbers and their sum
For all but one positive integer triplet (a; b; c) with a < b < c and b < 6, we decide whether there are algebraic numbers α,β and γ of degrees a, b and y, respectively, such that α+β+γ = 0. The undecided case (6; 6; 8) will be included in another paper. These results imply, for example, th...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2012 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:91341 |
| Acceso en línea: | https://ddd.uab.cat/record/91341 https://dx.doi.org/urn:doi:10.5565/PUBLMAT_56212_07 |
| Access Level: | acceso abierto |
| Palabra clave: | Algebraic number Sum-feasible Abc degree problem |
| Sumario: | For all but one positive integer triplet (a; b; c) with a < b < c and b < 6, we decide whether there are algebraic numbers α,β and γ of degrees a, b and y, respectively, such that α+β+γ = 0. The undecided case (6; 6; 8) will be included in another paper. These results imply, for example, that the sum of two algebraic numbers of degree 6 can be of degree 15 but cannot be of degree 10. We also show that if a positive integer triplet (a; b; c) satisfies a certain triangle-like inequality with respect to every prime number then there exist algebraic numbers α,β γ of degrees a, b, c such that α+β+γ = 0. We also solve a similar problem for all (a; b; c) with a < b < c and b <6 by finding for which a, b, c there exist number fields of degrees a and b such that their compositum has degree c. Further, we have some results on the multiplicative version of the first problem, asking for which triplets (a; b; c) there are algebraic numbers and α, β and γ of degrees a, b and c, respectively, such that αβγ = 1. |
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