TITLE:
Infinite Parametric Families of Irreducible Polynomials with a Prescribed Number of Complex Roots
AUTHORS:
Catalin Nitica, Viorel Nitica
KEYWORDS:
Írreducible Polynomial, Complex Roots, Real Roots, Galois Theory
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.9 No.1,
December
17,
2018
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 has exactly n-2kcomplex non-real roots
if n is even and has exactly n-2k-1complex 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=1and n odd prime.