Translator Disclaimer
2012 A degree problem for two algebraic numbers and their sum
Paulius Drungilas, Artūras Dubickas, Chris Smyth
Publ. Mat. 56(2): 413-448 (2012).

## Abstract

For all but one positive integer triplet $(a,b,c)$ with $a\leqslant b\leqslant c$ and $b\leqslant 6$, we decide whether there are algebraic numbers $\alpha$, $\beta$ and $\gamma$ of degrees $a$, $b$ and $c$, respectively, such that $\alpha+\beta+\gamma=0$. The undecided case $(6,6,8)$ will be included in another paper. These results imply, for example, that the sum of two algebraic numbers of degree $6$ can be of degree $15$ but cannot be of degree $10$. We also show that if a positive integer triplet $(a,b,c)$ satisfies a certain triangle-like inequality with respect to every prime number then there exist algebraic numbers $\alpha$, $\beta$, $\gamma$ of degrees $a$, $b$, $c$ such that $\alpha+\beta+\gamma=0$. We also solve a similar problem for all $(a,b,c)$ with $a\leqslant b\leqslant c$ and $b\leqslant 6$ by finding for which $a$, $b$, $c$ there exist number fields of degrees $a$ and $b$ such that their compositum has degree $c$. Further, we have some results on the multiplicative version of the first problem, asking for which triplets $(a,b,c)$ there are algebraic numbers $\alpha$, $\beta$ and $\gamma$ of degrees $a$, $b$ and $c$, respectively, such that $\alpha\beta\gamma=1$.

## Citation

Paulius Drungilas. Artūras Dubickas. Chris Smyth. "A degree problem for two algebraic numbers and their sum." Publ. Mat. 56 (2) 413 - 448, 2012.

## Information

Published: 2012
First available in Project Euclid: 19 June 2012

zbMATH: 1297.11133
MathSciNet: MR2978330

Subjects:
Primary: 11R04 , 11R32

Keywords: $abc$ degree problem , Algebraic number , sum-feasible