The product of a quartic and a sextic number cannot be octic (Q6595245)

A triplet \((a,b,c)\in \mathbb{N}^{3}\) is said to be product-feasible if there exist two algebraic numbers \(\alpha \) and \(\beta \) of degrees \(a\) and \(b\), respectively, such that the product \(\alpha \beta \) is of degree \(c\).\N\NIn the paper under review, the authors show that the triplet \((4,6,8)\) is not product-feasible, thus completing the classification of product-feasible triplets \((a,b,c)\) with \(a\leq b\leq \min \{7,c\}\). Also, they generally prove that for any integers \(n\geq 2\) and \(k\geq 1\), the triplet \((n,(n-1)k,nk)\) is product-feasible if and only if \(n\) is a prime number.