Infinite Parametric Families of Irreducible Polynomials with a Prescribed Number of Complex Roots - Open Journal of Discrete Mathematics

OJDM > Vol.9 No.1, January 2019

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

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DOI: 10.4236/ojdm.2019.91001 855 Downloads 1,637 Views

Author(s)

Catalin Nitica¹, Viorel Nitica²

Affiliation(s)

¹Technical College Dimitrie Leonida, Bucharest, Romania.
²Department of Mathematics, West Chester University, West Chester, USA.

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.

KEYWORDS

Írreducible Polynomial, Complex Roots, Real Roots, Galois Theory

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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