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

• Autorzy: Paulius Drungilas , Artūras Dubickas
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 12F05: Algebraic extensions
• 11R04: Algebraic numbers; rings of algebraic integers
• 11R32: Galois theory
• 20B35: Subgroups of symmetric groups

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2016
• Tom: 143
• Numer: 2
• Strony: 159-167

Daty

wydano

2016

Twórcy

autor

Paulius Drungilas

• Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania

autor

Artūras Dubickas

• Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
• Institute of Mathematics and Informatics, Vilnius University, Akademijos 4, Vilnius LT-08663, Lithuania

Identyfikatory

DOI

10.4064/cm6634-12-2015