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

Paulius Drungilas¹ ; Artūras Dubickas²

¹ Department of Mathematics and Informatics Vilnius University Naugarduko 24 Vilnius LT-03225, Lithuania
² Department of Mathematics and Informatics Vilnius University Naugarduko 24 Vilnius LT-03225, Lithuania and Institute of Mathematics and Informatics Vilnius University Akademijos 4 Vilnius LT-08663, Lithuania

Colloquium Mathematicum, Tome 143 (2016) no. 2, pp. 159-167

Résumé

Let $\alpha $, $\beta $ and $\gamma $ be algebraic numbers of respective degrees $a$, $b$ and $c$ over $\mathbb Q$ such that $\alpha + \beta + \gamma = 0$. We prove that there exist algebraic numbers $\alpha _1$, $\beta _1$ and $\gamma _1$ of the same respective degrees $a$, $b$ and $c$ over $\mathbb Q$ 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 $(a,b,c)\in \mathbb N^3$ 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 $KL$ 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.

Détail

DOI : 10.4064/cm6634-12-2015

Keywords: alpha beta gamma algebraic numbers respective degrees mathbb alpha beta gamma prove there exist algebraic numbers alpha beta gamma respective degrees mathbb alpha beta gamma proves previously formulated conjecture investigate problem describing set triplets mathbb which there exist finite field extensions fixed field degrees respectively degree compositum equals towards another earlier formulated conjecture under certain natural assumptions related inverse galois problem set triplets forms multiplicative semigroup