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 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.
Share and Cite:
Nitica, C. and Nitica, V. (2019) Infinite Parametric Families of Irreducible Polynomials with a Prescribed Number of Complex Roots.
Open Journal of Discrete Mathematics,
9, 1-6. doi:
10.4236/ojdm.2019.91001.
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