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

In this note, for any pair of natural numbers (n,k), n≥3, k≥1, and 2k, we construct an infinite family of irreducible polynomials of degree n, with integer coefficients, that has exactly n-2k？complex non-real roots if n is even and has 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.