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

@inproceedings{Drungilas2016OnDO, title={On degrees of three algebraic numbers with zero sum or unit product}, author={Paulius Drungilas and Artūras Dubickas}, year={2016} }

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 α1β1γ1 = 1. This proves a previously formulated conjecture. We also investigate the problem of describing the set of triplets (a, b, c) ∈ N 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…

## 3 Citations

### Multiplicative dependence of the translations of algebraic numbers

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In this paper, we first prove that given pairwise distinct algebraic numbers α 1 ,…,α n , the numbers α1+t,…,αn+tare multiplicatively independent for all sufficiently large integers t. Then, for a…

### A degree problem for the compositum of two number fields*

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The triplet (a, b, c) of positive integers is said to be compositum-feasible if there exist number fields K and L of degrees a and b, respectively, such that the degree of their compositum KL equals…

### Multiplicative dependence and independence of the translations of algebraic numbers

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In this paper, we first prove that given pairwise distinct algebraic numbers $\alpha_1, \ldots, \alpha_n$, the numbers $\alpha_1+t, \ldots, \alpha_n+t$ are multiplicatively independent for all…

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