9134120141007154704.0artpubuabdriveroai:ddd.uab.cat:91341doi10.5565/PUBLMAT_56212_07articleid02141493v56n2p413engDrungilas, P.A degree problem for two algebraic numbers and their sumFor 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.info:eu-repo/semantics/restrictedAccessTots els drets reservatshttp://www.europeana.eu/rights/rr-f/AnglèsrecercaAlgebraic numberSum-feasibleAbc degree probleminfo:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionDubickas, A.Smyth, C.Vol. 56, Núm. 2 (2012), p. 413-448Publicacions matemàtiques0214-149336452090http://ddd.uab.cat/pub/pubmat/pubmat_a2012v56n2/pubmat_a2012v56n2p413.pdf0413448256pubmat_a2012v56n22012ARTPUBPUBMATUABfile0MD57885f09072f3a3888ef0b77ccc7dfd58452090bytestream1.5filepathpub/pubmat/pubmat_a2012v56n2/pubmat_a2012v56n2p413.pdfdisk