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