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

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Document pages: 6 pages

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 integercoefficients, that has exactly n-2k complex non-real rootsif n is even and has exactly n-2k-1 complex non-real rootsif n is odd. Our work generalizes atechnical result of R. Bauer, presented in the classical monograph “BasicAlgebra” of N. Jacobson. It is used there to construct polynomials with Galoisgroups, the symmetric group. Bauer’s result covers the case k=1 and n odd prime.

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