Reducibility of lacunary polynomials, VII

Abstract

Letk>1 and let$a_j (0 \leqslant j \leqslant k)$ be non-zero algebraic numbers contained in the field$\mathbb{K}_0 = \mathbb{Q}(a_1 /a_0 ,...,a_k /a_0 )$. It is shown that for almost all, in the sense of density integer vectorsn ₁,...,n _k the polynomial$a_0 + \sum\limits_{j = 1}^k {a_j x^{n_j } }$ becomes irreducible over$\mathbb{K}_0$ on dividing by the product of all factorsx−ξ, where ξ is a root of unity.