Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
134 tokens/sec
GPT-4o
10 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

An extension of Schur's irreducibility result (2305.04781v1)

Published 8 May 2023 in math.NT

Abstract: Let $n\geq 2$ be an integer. Let $\phi(x)$ belonging to $\mathbb{Z}[x]$ be a monic polynomial which is irreducible modulo all primes less than or equal to $n$. Let $a_0(x), a_1(x), \dots, a_{n-1}(x)$ belonging to $\mathbb{Z}[x]$ be polynomials each having degree less than $\deg \phi(x)$ and $a_n$ be an integer. Assume that $a_n$ and the content of $a_0(x)$ are coprime with $n!$. In the present paper, we prove that the polynomial $\sum\limits_{i=0}{n-1} a_i(x)\frac{\phi(x)i}{i!}+a_n\frac{\phi(x)n}{n!}$ is irreducible over the field $\mathbb{Q}$ of rational numbers. This generalizes a well known result of Schur which states that the polynomial $\sum\limits_{i=0}{n} a_i\frac{xi}{i!}$ is irreducible over $\mathbb{Q}$ for all $n\geq 1$ when each $a_i\in \mathbb{Z}$ and $|a_0|=|a_n|=1$. The present paper also extends a result of Filaseta thereby leading to a generalization of the classical Sch\"{o}nemann Irreducibility Criterion.

Summary

We haven't generated a summary for this paper yet.