| Title |
On degrees of three algebraic numbers with zero sum or unit product |
| Authors |
Drungilas, Paulius ; Dubickas, Artūras |
| DOI |
10.4064/cm6634-12-2015 |
| Full Text |
|
| Is Part of |
Colloquium Mathematicae.. Warsaw : Polska Akademia Nauk. 2016, Vol. 143, No. 2, p. 159-167.. ISSN 0010-1354. eISSN 1730-6302 |
| Keywords [eng] |
algebraic number ; sum-feasible ; product-feasible ; compositum-feasible ; inverse Galois problem ; semigroup |
| Abstract [eng] |
Let alpha, beta and gamma be algebraic numbers of respective degrees a, b and c over Q such that alpha vertical bar beta vertical bar gamma = 0. We prove that there exist algebraic numbers alpha(1), beta(1) and gamma(1) of the same respective degrees a, b and c over Q such that alpha(1)beta(1)gamma(1) = 1. This proves a previously formulated conjecture. We also investigate the problem of describing the set of triplets (a, b, c) is an element of N-3 for which there exist finite field extensions K/k and L/k (of a fixed field k) of degrees a and b, respectively, such that the degree of the compositum K L over k equals c. Towards another earlier formulated conjecture, under certain natural assumptions (related to the inverse Galois problem), we show that the set of such triplets forms a multiplicative semigroup. |
| Published |
Warsaw : Polska Akademia Nauk |
| Type |
Journal article |
| Language |
English |
| Publication date |
2016 |
| CC license |
|
