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 ) ∈ N 3 \left(a,b,c)\in {{\mathbb{N}}}^{3} with a ≤ b ≤ c a\le b\le c and b ≤ 7 b\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 , b a,b , and c c over Q {\mathbb{Q}} , respectively, such that α β = 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 ≥ 2 n\ge 2 and k ≥ 1 k\ge 1 , the triplet ( a , b , c ) = ( n , ( n − 1 ) k , n k ) \left(a,b,c)=\left(n,\left(n-1)k,nk) is product-feasible if and only if n n 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.