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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm6634-12-2015,
author = {Paulius Drungilas and Art\=uras Dubickas},
title = {On degrees of three algebraic numbers with zero sum or unit product},
journal = {Colloquium Mathematicae},
volume = {144},
year = {2016},
pages = {159-167},
zbl = {06574979},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm6634-12-2015}
}
Paulius Drungilas; Artūras Dubickas. On degrees of three algebraic numbers with zero sum or unit product. Colloquium Mathematicae, Tome 144 (2016) pp. 159-167. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm6634-12-2015/