A Degree Problem for Two Algebraic Numbers and Their Sum

  • PAULIUS DRUNGILAS
  • Published 2011

Abstract

For all but one positive integer triplet (a, b, c) with a 6 b 6 c and b 6 6, we decide whether there are algebraic numbers α, β and γ of degrees a, b and c, 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 6 b 6 c and b 6 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 α, β and γ of degrees a, b and c, respectively, such that αβγ = 1.

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Cite this paper

@inproceedings{DRUNGILAS2011ADP, title={A Degree Problem for Two Algebraic Numbers and Their Sum}, author={PAULIUS DRUNGILAS}, year={2011} }