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## Colloquium Mathematicum

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

### Tom 143 / 2016

Colloquium Mathematicum 143 (2016), 159-167 MSC: Primary 11R04; Secondary 11R32, 12F05, 20B35. DOI: 10.4064/cm6634-12-2015 Opublikowany online: 3 December 2015

#### Streszczenie

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.

#### Autorzy

• Paulius DrungilasDepartment of Mathematics
and Informatics
Vilnius University
Naugarduko 24
Vilnius LT-03225, Lithuania
e-mail
• Artūras DubickasDepartment of Mathematics and Informatics
Vilnius University
Naugarduko 24
Vilnius LT-03225, Lithuania
and
Institute of Mathematics and Informatics
Vilnius University