91341
20151128213201.0
doi
10.5565/PUBLMAT_56212_07
artpubuab
artpubuabnoredi
driver
oai:ddd.uab.cat:91341
articleid
02141493v56n2p413
eng
Drungilas, P.
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, 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/openAccess
Tots els drets reservats
http://www.europeana.eu/rights/rr-f/
Anglès
recerca
Algebraic number
Sum-feasible
Abc degree problem
info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
Dubickas, A.
Smyth, C.
Vol. 56, Núm. 2 (2012), p. 413-448
Publicacions matemàtiques
0214-1493
36
452090
http://ddd.uab.cat/pub/pubmat/pubmat_a2012v56n2/pubmat_a2012v56n2p413.pdf
0413
448
2
56
pubmat_a2012v56n2
2012
ARTPUB
PUBMAT
UAB