2023/05/08 by Ankita Jindal, Jindal, Ankita, Sudesh K. Khanduja +1
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2305.04781
openalex publication_date 2023/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let n≥ 2 be an integer. Let ϕ(x) belonging to ℤ[x] be a monic polynomial which is irreducible modulo all primes less than or equal to n. Let a0(x), a1(x), …, an-1(x) belonging to ℤ[x] be polynomials each having degree less than °ϕ(x) and an be an integer. Assume that an and the content of a0(x) are coprime with n!. In the present paper, we prove that the polynomial ∑i=0n-1 ai(x)(ϕ(x)i)/(i!)+an(ϕ(x)n)/(n!) is irreducible over the field ℚ of rational numbers. This generalizes a well known result of Schur which states that the polynomial ∑i=0n ai(xi)/(i!) is irreducible over ℚ for all n≥ 1 when each ai∈ ℤ and |a0|=|an|=1. The present paper also extends a result of Filaseta thereby leading to a generalization of the classical Schönemann Irreducibility Criterion.