eng
A degree problem for two algebraic numbers and their sum
Drungilas, P.
Dubickas, A.
Smyth, C.
info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
2012
Algebraic number
Sum-feasible
Abc degree problem
Publicacions matemàtiques ; Vol. 56, Núm. 2 (2012), p. 413-448
http://ddd.uab.cat/record/91341
10.5565/PUBLMAT_56212_07
oai:ddd.uab.cat:91341
02141493v56n2p413
info:eu-repo/semantics/openAccess
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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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