| Abstract [eng] |
In this article, we prove that the product of two algebraic numbers of degrees 4 and 6 over ℚ cannot be of degree 8. This completes the classification of so-called product-feasible triplets (a, b, c) ∈ ℕ3 with a ≤ b ≤ c and b ≤ 7. The triplet (a, b, c) is called product-feasible if there are algebraic numbers α, β, and γ of degrees a, b, and c over ℚ, respectively, such that αβ = γ. In the proof, we use a proposition that describes all monic quartic irreducible polynomials in ℚ[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 and k ≥ 1, the triplet (a, b, c) = (n, (n − 1)k, nk) is product-feasible if and only if n is a prime number. The choice (n, k) = (4, 2) recovers the case (a, b, c) = (4, 6, 8) as well. |
