On degrees of three algebraic numbers with zero sum or unit product

Paulius Drungilas, and Artūras Dubickas. "On degrees of three algebraic numbers with zero sum or unit product." Colloquium Mathematicae 143.2 (2016): 159-167. <http://eudml.org/doc/283561>.

@article{PauliusDrungilas2016,
abstract = {Let α, β and γ be algebraic numbers of respective degrees a, b and c over ℚ such that α + β + γ = 0. We prove that there exist algebraic numbers α₁, β₁ and γ₁ of the same respective degrees a, b and c over ℚ such that α₁ β₁ γ₁ = 1. This proves a previously formulated conjecture. We also investigate the problem of describing the set of triplets (a,b,c) ∈ ℕ³ 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 KL 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.},
author = {Paulius Drungilas, Artūras Dubickas},
journal = {Colloquium Mathematicae},
keywords = {algebraic number; sum-feasible; product-feasible; compositumfeasible; inverse Galois problem; semigroup},
language = {eng},
number = {2},
pages = {159-167},
title = {On degrees of three algebraic numbers with zero sum or unit product},
url = {http://eudml.org/doc/283561},
volume = {143},
year = {2016},
}

TY - JOUR
AU - Paulius Drungilas
AU - Artūras Dubickas
TI - On degrees of three algebraic numbers with zero sum or unit product
JO - Colloquium Mathematicae
PY - 2016
VL - 143
IS - 2
SP - 159
EP - 167
AB - Let α, β and γ be algebraic numbers of respective degrees a, b and c over ℚ such that α + β + γ = 0. We prove that there exist algebraic numbers α₁, β₁ and γ₁ of the same respective degrees a, b and c over ℚ such that α₁ β₁ γ₁ = 1. This proves a previously formulated conjecture. We also investigate the problem of describing the set of triplets (a,b,c) ∈ ℕ³ 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 KL 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.
LA - eng
KW - algebraic number; sum-feasible; product-feasible; compositumfeasible; inverse Galois problem; semigroup
UR - http://eudml.org/doc/283561
ER -