An extension of a second irreducibility theorem of I. Schur

2023/05/28 by Jakhar, Anuj, Kalwaniya, Ravi
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2306.03294

Abstract

Let n ≠ 8 be a positive integer such that n+1 ≠ 2^u for any integer u≥ 2. Let ϕ(x) belonging to ℤ[x] be a monic polynomial which is irreducible modulo all primes less than or equal to n+1. Let a_j(x) with 0≤ j≤ n-1 belonging to ℤ[x] be polynomials having degree less than degϕ(x). Assume that the content of (a_na₀(x)) is not divisible by any prime less than or equal to n+1. In this paper, we prove that the polynomial f(x) = a_n(ϕ(x)ⁿ)/((n+1)!)+ ∑_j=0^n-1a_j(x)\fracϕ(x)^j(j+1)! is irreducible over the field ℚ of rational numbers. This generalises a well-known result of Schur which states that the polynomial ∑_j=0ⁿa_j\fracx^j(j+1)! with a_j ∈ ℤ and |a₀| = |a_n| = 1 is irreducible over ℚ. We illustrate our result through examples.

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