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Home > Articles > Published articles > A degree problem for two algebraic numbers and their sum 
Date:  2012 
Abstract:  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 trianglelike 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. 
Rights:  Tots els drets reservats 
Language:  Anglès 
Document:  article ; recerca ; publishedVersion 
Subject:  Algebraic number ; Sumfeasible ; Abc degree problem 
Published in:  Publicacions matemàtiques, Vol. 56, Núm. 2 (2012) , p. 413448, ISSN 02141493 
36 p, 441.5 KB 
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Articles > Published articles > Publicacions matemàtiques