A degree problem for two algebraic numbers and their sum

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 $\alpha\beta\gamma=1$.