Infinite Parametric Families of Irreducible Polynomials with a Prescribed Number of Complex Roots

Abstract

In this note, for any pair of natural numbers (n,k) , n≥3 , k≥1 , and 2k<n , we construct an infinite family of irreducible polynomials of degree n , with integer coefficients , that ha s exactly n-2k complex non-real roots if n is even and ha s exactly n-2k-1 complex non-real roots if n is odd. Our work generalizes a technical result of R. Bauer, presented in the classical monograph “Basic Algebra” of N. Jacobson. It is used there to construct polynomials with Galois groups , the symmetric group. Bauer’s result covers the case k=1 and n odd prime.