The product of a quartic and a sextic number cannot be octic

Abstract

In this article, we prove that the product of two algebraic numbers of degrees 4 and 6 over Q{\mathbb{Q}} cannot be of degree 8. This completes the classification of so-called product-feasible triplets (a,b,c)∈N3\left(a,b,c)\in {{\mathbb{N}}}^{3} with a≤b≤ca\le b\le c and b≤7b\le 7. The triplet (a,b,c)\left(a,b,c) is called product-feasible if there are algebraic numbers α,β\alpha ,\beta , and γ\gamma of degrees a,ba,b, and cc over Q{\mathbb{Q}}, respectively, such that αβ=γ\alpha \beta =\gamma . In the proof, we use a proposition that describes all monic quartic irreducible polynomials in Q[x]{\mathbb{Q}}\left[x] with four roots of equal moduli and is of independent interest. We also prove a more general statement, which asserts that for any integers n≥2n\ge 2 and k≥1k\ge 1, the triplet (a,b,c)=(n,(n−1)k,nk)\left(a,b,c)=\left(n,\left(n-1)k,nk) is product-feasible if and only if nn is a prime number. The choice (n,k)=(4,2)\left(n,k)=\left(4,2) recovers the case (a,b,c)=(4,6,8)\left(a,b,c)=\left(4,6,8) as well.

Keywords

• algebraic numbers
• product-feasible triplets
• field extensions
• 11r04
• 11r21
• 11r32
• 12f10